(For both scholars and laypersons.)
Ayn Rand's philosophy, which she called Objectivism, has received widely varied reception, from "It's the key to solving our crises of civilization" all the way to "It's a pseudo-philosophy not worth taking seriously." (Disclosure: My own view is much, much closer to the former than the latter, with some reservations and qualifications.) This guide is meant to convey what a serious scholar (or "student of Objectivism") might come away with after a careful study of Rand's ideas. I will go over Rand's views on philosophy, branch by branch, and indicate what strengths (and, in some instances, weaknesses) a student of philosophy can expect to find therein.
First, Rand summed up her philosophy as follows: "My philosophy, in essence, is the concept of man as a heroic being, with his own happiness as the moral purpose of his life, with productive achievement as his noblest activity, and reason as his only absolute."
(http://aynrandlexicon.com/lexicon/objectivism.html)
Now, to see how this bears on subject matter in philosophy:
Metaphysics
Rand's views here are not much advanced beyond the ancient Greek - namely, and especially, Aristotelian - conception of the world as existing independent of us and ordered according to "natural law." Her axioms of existence, identity and consciousness are fundamental-level identifications implicit within all other statements, and are denied upon pain of contradiction. Her neo-Aristotelian metaphysics might best be identified as a standard statement of "classical realism" - the view of the world essentially contained in "common sense": we come to the world without any power to alter or construct it; it is absolute and unyielding; it exhibits causal regularity to which we have to conform to achieve any kind of cognitive or practical success.
Rand's views on the subject of God are "hard-line" and philosophically controversial. The arguments offered by Rand and others on this are unlikely to convince believers or theologians. They amount in effect to the claim that the universe - which is considered synonymous with "existence" or the totality of all that exists - could not come into or go out of existence, and that traditional talk of God leads to a negation of a rational understanding of the world. In an interview on the Donahue show, Rand made a comment that "There is no such thing as a disorderly universe," the implication being that there is no need to posit God to explain the order in the world. While the universe being orderly of necessity falls out of Rand's classical-realist metaphysics, and its being orderly needn't require an order-er or creator or designer, it's far from clear how this rules out God's existence by Rand's hard-line reasoning.
Rand did not get into the subject of God beyond some basic metaphysical claims; she did not talk about the Problem of Evil, for instance. Neither did she talk about God, much, period: it wasn't a subject of interest to her in itself, apart from the historical-cultural phenomenon of belief in such a deity or higher power. There do appear to be two distinct strands offered, however: the usual "there's no compelling proof" one shared by a whole host of non-believers, on the one hand, and the hard-line metaphysical one in which God's existence implies a contradiction or rejection of the axioms, on the other.
Rand rejects the "materialist" label, where that means being committed to the view that all reality is ultimately material; she associated such a label with "vulgar materialism" which denies an irreducible reality and causal efficacy to conscious processes. It is hard to escape this label, however, if materialism amounts only to the view that existence exists independently of consciousness, per classical or "common sense" realism. Rand is of the view that existence existing independent of consciousness implies that there can't be a God, which is presumably defined such that it is a conscious entity that exists ontologically prior to the world. This seems to miss the claim of theologians that God is at once existing and conscious (just like us), without any claim that God's consciousness is somehow prior to existence. It is unclear from her or her designated spokespersons' extant written arguments how the axioms can be invoked to handle challenges like this one.
In sum, I don't think there's much to be gotten out of Objectivist metaphysics than some pretty standard classical-realist claims and some strange-looking arguments against God's existence and against other things which Rand and others claim violates the axioms. We can already appeal to Aristotelian or other arguments in favor of classical realism and get much the same idea about the world and our relation to it.
Epistemology
To understand Rand's epistemological project, it helps first to get clear on how she conceived that project. Her aim was fundamentally a practical one and a methodological one. The practical aim was to provide a basic guide for ordering our thought processes in a way that makes us most efficacious in living. The methodological aim is the spelling-out of the practical one: defining the methods by which we best order our cognitive processes.
Rand regarded her theory of concepts as her most significant philosophic achievement. She aimed to provide not just a solution to the traditional "problem of universals," but also to provide a fundamental accounting of how we think about anything and everything, from the most abstract to the most particular, to the most theoretical to the most mundane. Our knowledge is held in the form of concepts, and concepts are integrations of percepts (the perceptually-given). If we can provide a systematic accounting for how thought, properly done, is carried out, then we've done all the epistemological work Rand is concerned about.
Rand spent some 20 years, prior to laying out her theory of concepts in Introduction to Objectivist Epistemology, testing her ideas about the roots of our concepts. Her theory in essence is that human cognition involves measurement (of particulars) and measurement-omission (in forming concepts, i.e., by integrating percepts into mental units). All abstractions are ultimately tied to the perceptual, either in simpler and direct terms (via lower-order concepts, e.g., concepts of color), or in more complex and indirect terms (involving higher-order concepts, e.g., "love" and "justice"). The higher-order abstractions require definition in terms of the lower-order ones.
Chains of reasoning, if they are to be considered sound, depend ultimately on reducibility to particulars. This requires keeping our concepts well-organized within a hierarchy, which establishes the context for any concepts we employ. We should be able to relate, i.e., to integrate any item of knowledge to all other items of knowledge within the hierarchy. Concepts well-formed and well-organized serve a crucial practical need, namely, mental unit-economy and, in turn, mental and practical efficacy. Concepts and definitions are condensations of vast bodies of observation, with the particular measurements as they apply to concrete data omitted from these condensations. Rand treats of "borderline cases" (as, say, between "chair" and "sofa") by stressing the need for a mind to organize its contents by the most efficacious means available; that may or may not require forming new concepts to handle "borderline" cases. The basic determinant for forming a concept or definition of anything is to identify the fundamental similarities among concretes, i.e., the similarities which (existentially or metaphysically) make the greatest number of similarities possible, and which (epistemologically) explains the greatest number of other similarities (These considerations should be guided by Rand's "Razor", which states that concepts should be neither expanded nor reduced beyond necessity.)
The basic cognitive method Rand endorsed and applied here is one of keeping all one's mental contents very well-organized and at least fairly readily reducible to the perceptual. Doing all this is not just a mental exercise but, rather, serves a crucial life need. Her chief aim was to make consciousness as efficacious as possible in dealing with everyday problems. In that regard, Rand did not particularly concern herself with a number of "background" issues in epistemology that have concerned a great deal of other philosophers. If they approach Rand with an eye to these "background" issues, they might not find much to sink their teeth into. An Objectivist, meanwhile, will maintain that any discussion of any issues, "background" or otherwise, requires the use of well-formed and well-applied concepts, and that the relevance of these background issues must be explained in terms of relevance and use for daily life; otherwise, it is idle speculation.
So, to take, for instance, the "problem of necessity" addressed by Hume and Kant, it needs to be explained how this is an issue that should be of concern to us. There is a "pragmatic" attitude involved here: if we already have a well-ordered system of concepts that aids us in our daily lives, then it becomes some other kind of concern (a "theoretical" one) whether the concepts we have reflects an inherent necessity in nature. Even if we don't have a "satisfactory grounding of necessity," i.e., one satisfying to all but skeptics, we still have to get on with the task of living. At the same time, any discussion of the "problem of necessity" should presuppose that we're employing our concepts in such a discussion in well-grounded (i.e., perceptually-grounded) ways.
If all knowledge is grounded in the perceptual, according to Rand, then the label "empiricist" might seem to fit. To be sure, Rand, despite her advocacy of reason, did not identify with the "rationalist" tradition; in fact, she saw in rationalism a tendency to treat ideas or concepts in a way that detaches them from their proper grounding in the perceptual - and thereby to sever philosophy from the needs of life. However, to lump her in with Hume using the "empiricist" label is to ignore the basic difference between them: the task of the cognitive faculty (which Rand called "reason") in Rand's philosophy is to integrate sense data into a non-contradictory whole, even if "only" for practical purposes; Hume, as abstract theorist, was more interested in the "problem of necessity," while he's often seen (fairly or not) as a leading advocate against the idea that we can rationally or objectively integrate percepts. In any event, Rand's perception-based theory of knowledge is more in the spirit of Aristotle than of Hume.
Ethics
Rand is perhaps most famous for her advocacy of ethical egoism, or as she referred to it, a rational selfishness or morality of self-interest. Evident from a careful reading of her arguments, though, is that she is not an advocate of these things as typically or widely understood.
Her basic point about ethics - "a code of values to guide man’s choices and actions - the choices and actions that determine the purpose and the course of his life" - is that it is a guide to living well, or living happily. In her understanding of things, that makes any good code of ethics egoistic by definition: living well or happily just means to be living self-interestedly. Insofar as Rand's "ethical egoism" is damned or rejected by moral theorists, they also damn or reject the ancient conception of ethics as, likewise, being concerned with the task of living well or happily. Objectivist ethics is hardly anything more or other than an updating of Aristotelian eudaemonist ethics.
Rand offers a neo-Aristotelian argument on the foundations of the concept of "value" or "goodness"; she locates value-significance in living phenomena, and a narrower sub-division - the moral - is concerned with the achievement of what is of value through the exercise of choice. She shares with Aristotle the basic conviction that living well as humans means living rationally and intelligently - that rational and intelligent living is our best (perhaps only) guarantee to achieving a stable and enduring happy livelihood. The basic form of right living for both Aristotle and Rand is virtue, the integral commitment to living as reason requires. For Rand, the basic virtue, which explains all the other virtues, is the virtue of rationality. Rationality is the locus of the interaction between her epistemology and her ethics, or between thought and practice. Humans are distinguished by their mode of functioning - most fundamentally, the exercise of reason. If we perfect the use of our cognitive faculties, that is our best and only means to perfecting our lives as a whole.
In the contemporary professional literature, eudaemonism generally is discussed in David L. Norton's Personal Destinies: A Philosophy of Ethical Individualism (Princeton, 1976), while Rand's eudaemonism more specifically is discussed in Tara Smith's Ayn Rand's Normative Ethics: The Virtuous Egoist (Cambridge, 2006). Rand's ethics - again, best understood as a version of neo-Aristotelian eudaemonism - is the part of her philosophy which has seen the most coverage and ground gained in the philosophy profession. (Her epistemology, by contrast, has received hardly any treatment, negative or positive.) The egoism she espoused is best understood in connection with the basic summary statement near the beginning of this posting: the idea that achievement of happiness is the moral purpose of one's life, and that we should strive to be "heroic" or great in our lives. Arguably this fits right in with an Aristotelian version of eudaemonism.
The strength of Rand's ethics depends on the strength of eudaemonist ethics as such; eudaemonism has become a force to be reckoned with in contemporary moral theories, and as such moral theorists ignore it and Rand at their peril.
Rand's meta-ethics - her grounding of the concept of "value" in the concept of "life" - has been much the subject of analysis and criticism, and much of that discussion centers on whether Rand (or any other thinker) successfully derives an "ought" from an "is." Rand contours her claims about living things so as not to commit herself to defunct teleological doctrines, but her claims can still be challenged on grounds of whether the functions of living systems are ordered for "the preservation of the organism's own life" (as distinct from or perhaps inclusive of reproduction), or on whether we can get a clear picture of functional organization that gives us the kind of ethical views we usually find plausible. Again, her views here are closely related to Aristotle's, and are also echoed in such neo-Aristotelians as Philippa Foot (Natural Goodness, Oxford, 2001). Her views in these areas seem to be well-reflected in the mainstream literature and the tradition, and as such are more or less right in the thick of things as contemporary meta-ethics is concerned.
Political Philosophy
Rand's politics in essence is an extension of her ethics: she is an individualist, and in an individualist ethos the basic function of government is to secure the conditions - rights - under which people can pursue their happiness through the exercise of their own minds. Her views have predecessors in Locke, Jefferson and Spencer, but her explanation of the relation between ethics and politics is rather original if not right on target. Her basic identification here is that force and mind are opposites. If eudaemonia is necessarily rationally-directed activity, then we require the freedom to exercise our own minds and judgments to achieve it. Further, the propriety of pursuing happiness grounds the right to pursue happiness. Rights - the basic concept in social and political philosophy - are principles "defining and sanctioning a man's freedom of action in a social context." Rights define freedoms rather than specific objects to which we might lay claim. (Property rights, as an extension of moral personhood, specify freedoms as to the disposition of goods, not rights to goods themselves.)
Rand's substantive conclusions about rights are closely echoed in Robert Nozick's Anarchy, State, and Utopia (Basic Books, 1974), and both Rand and Nozick stand opposed to the contemporary "liberal" mainstream, well-represented by John Rawls's A Theory of Justice (Harvard, 1971). For Rawlsians and other liberals, much hinges on whether a Randian or Nozickian conception of justice reliably serves, in practice, the legitimate interests of "the less well-off." In fact, Rand's theory, especially, would merit universal assent only if it could be addressed to the rational self-interest of all members of society; this is a natural consequence of Rand's universalization of egoistic norms (as opposed to the caricature of pretty much all versions of egoism as exception-making on behalf of a talented few). This might "force" Rand and/or Nozick to resort to "empirical" or social-scientific claims on behalf of their laissez-faire capitalist conception of rights and justice.
There are, however, at least two considerations a Randian can raise in addition to acknowledging whatever is the case social-scientifically: (1) Even if there are good arguments showing that people should adopt certain measures aimed at improving the condition or life-prospects of the "less well-off," it doesn't follow that this should be done via the State, an apparatus of coercion; and (2) There is more to the story than simply justice in a political sense: Rand would share with a number of critics of Rawls (or of contemporary mainstream liberalism) the view that a flourishing society would require a conception of virtue for people to adopt; for Rand, especially, a society the members of which are educated in the ways of virtue would be one where politics is hardly needed at all (except for the most minimal rights-protecting functions), for such a society would likely be very full of people who are already flourishing. (A big question, then, is how we could educate a large segment of society in the ways of virtue, i.e., what sort of program of education we ought and are able to adopt and apply effectively. In any event, this process of education would require time.)
Aesthetics
Rand's aesthetics have been little-explored, including by the current author. Here some basics will have to do: Rand saw art as a means of depicting life "as it might be and ought to be." Art serves a fundamental spiritual need in man, and its products affect the audience on a sense-of-life level, sense of life being "a pre-conceptual equivalent of metaphysics, an emotional, subconsciously integrated appraisal of man and of existence." Aesthetics, then, would be concerned with the relation between consciously and rationally-adopted values on the one hand, and the immediate, emotional, and subconscious reaction to works of art or beauty on the other. Further, the spiritual needs of man are as integral to Rand's worldview as they are to any religious one; they are located here in the natural world rather than in relation to a supernatural one.
Other
In connection with both epistemology and "sense of life," Rand and other Objectivists have spoken of the subject of psycho-epistemology, or "the study of man’s cognitive processes from the aspect of the interaction between the conscious mind and the automatic functions of the subconscious." This is an intriguing line of study, for it informs us on how people habitually approach their mental content. Psycho-epistemology is chiefly concerned with method, from the standpoint of how our rationally and consciously-directed processes interact with the immediately and automatically given, such as emotions, subconsciously-given intuition, or even habituated (and therefore automatized) thought processes themselves. Automatization is a big concept here and arguably requires some development; among other things, it ties into how we understand the subconscious as being a repository of automatized content and method. A study of psychology also comes to bear on understanding what is automatized in our thought processes, and how. Rand's aim as epistemologist was to make our consciously-directed thought processes so well-formed and habituated that they work in harmony with the subconscious; ideally, sound thought processes would come more or less automatically after habituation - though always, of course, subject to volitional assessment.
Rand's "style" of doing philosophy as it pertains to polemics directed against other thinkers, leaves something to be desired. Many of her claims reduce other thinkers such as Kant to caricatures, and her approach is one of "good guys vs. villains" instead of one of acknowledging other philosophers as providing incomplete, perhaps confused, perhaps even bad, but nonetheless thoughtful and honestly-reasoned perspectives on the Big Issues. This is one aspect of Rand's approach that can be fairly called a blind spot - indeed, one that is often incomplete, confused, bad, etc., even if thoughtful or reasoned in some way or other. She should not be looked to for information on the history of philosophy, just as few commentators should be looked to for information on Rand's ideas herself; the best thing to do in both cases is to study the relevant literature. (It is fair to say, though, that she had a correct basic grasp of the importance of philosophy as it affects the course of history.) It is fair enough to say that Rand's basic sympathies were with Aristotle, while legitimate differences with Plato, Kant and others would arise from that basic sympathy.
Rand's foremost student, Leonard Peikoff, received her full endorsement in a general letter of recommendation (reproduced in Letters of Ayn Rand, Dutton, 1995), and his lecture courses on Objectivism (e.g. Understanding Objectivism, 1983-4) provide valuable insights into Objectivism especially in terms of methodology as distinct from content. The basic content of her philosophy is set forth in complete form in his Objectivism: The Philosophy of Ayn Rand (Dutton, 1991), which is based on a 1976 course which she endorsed. His later courses presuppose a familiarity with the content, and seek instead to provide a guide to everyday thinking processes (philosophically or non), stressing the central methodological concepts of integration, hierarchy, and context. While the courses are almost prohibitively expensive, and Peikoff adopts at times Rand's tendencies in polemics, a full scholarly study of Objectivism should take these courses into account. The aim with the courses, as with Rand's own writings, is the integration of theory and practice, of philosophy and life.
Conclusion
Rand is certainly worth taking seriously as a thinker, despite misgivings I and others have over certain things. I would not expect scholars to find a whole lot that is both new and compelling about her metaphysics. Her epistemology, if correctly understood in its aims, provides a lot of useful material in terms of thinking methods and practical application to the task of day-to-day living. A very small number of thinkers have even yet approached her epistemological writings in these terms. Her ethics stands or falls with the strength of eudaemonism, which is a very promising mainstream alternative to deontology and consequentialism. Her meta-ethics, dealing with concepts of goodness and value, should at the very least stimulate thoughts in these areas, and may help lead to fully-worked-out treatments by scholars of the concept of goodness in naturalistic or perceptually-based terms. Her politics stresses the fundamental importance of freedom of the individual and the tie between ethics and politics. In sum, her epistemology, ethics and political philosophy are areas of strength in her thinking, and where further scholarly study or detailed working-out is warranted.