Axioms

An axiom is a primary truth: an identification so fundamental that it cannot be reduced to anything simpler, cannot be proved because proof presupposes it, and cannot be denied because denial presupposes it too. Objectivism holds three — existence exists, A is A, and consciousness is conscious of something — and treats them as the base of the whole philosophy.

“An axiomatic concept is the identification of a primary fact of reality, which cannot be analyzed, i.e., reduced to other facts or broken into component parts.”
~ Ayn Rand, "Axiomatic Concepts", Introduction to Objectivist Epistemology

The three axioms

Existence exists. Something is. This is presupposed by every statement about anything, including the statement that nothing exists.
Identity: A is A. To exist is to be something in particular, with a definite nature. Aristotle's law of identity, which Rand restated as "existence is identity". See A is A and Law of Identity.
Consciousness is conscious. To be aware is to be aware of something. Consciousness is not a self-contained realm but a faculty of grasping what exists, which is why the axiom of consciousness presupposes the axiom of existence and not the reverse. See Axiom of Consciousness and Primacy of Existence.

Rand also treated a fourth identification as axiomatic in the field of ethics: that man possesses volition, the choice to think or not. See Axiom of Self-Consciousness.

Why an axiom cannot be proved — and does not need to be

The usual objection is that a starting point accepted without proof is arbitrary. The Objectivist answer is that an axiom is not accepted without evidence; it is accepted on the evidence of everything.

They are perceptually self-evident. An axiom is not deduced from prior premises but directly given in every act of perception. Looking at anything at all is simultaneously an instance of something existing, being what it is, and being perceived. See Perception.
Proof would be circular. Proof is the derivation of a conclusion from prior knowledge. There is no knowledge prior to the axioms, so demanding proof of them is demanding that they be derived from themselves.
They are validated by reaffirmation through denial. This is the distinctive Objectivist argument. Any attempt to deny an axiom must use it. A man who says "existence does not exist" is an existent using a consciousness to make a claim about reality. The denial does not fail because it is false but because it is self-eliminating — it destroys the conditions under which any assertion could mean anything. Attempting it is the stolen concept fallacy in its purest form.

Axiomatic concepts, not just axiomatic propositions

Rand's contribution in Introduction to Objectivist Epistemology was to locate the primaries one level below the propositions. "Existence exists" is a sentence; what it identifies is the axiomatic concept of existence.

An axiomatic concept has an unusual property: it cannot be defined in the ordinary way. A definition specifies a genus and a differentia — a wider category and what distinguishes the units within it. There is no category wider than existence, and nothing outside it to contrast it with. Axiomatic concepts are therefore defined ostensively, by pointing: that — and everything else. See Definitions and Fundamental.

They are also implicit in the knowledge of an infant long before he has words for them. A child grasping that a rattle is a rattle and not a blanket is using identity without being able to state it.

Common misunderstandings

That axioms are assumptions. An assumption is something one could coherently doubt and has chosen not to. The axioms cannot coherently be doubted, because doubting is itself an act of a consciousness about existents.
That they are like mathematical axioms. A geometrical axiom is a starting premise within a system, and alternative systems can begin from different ones. Philosophical axioms are not chosen; they identify what any system whatever must already rest on.
That "existence exists" is empty. It is deliberately contentless about what exists — that is the work of science. Its function is to rule out the alternatives: that reality is created by consciousness, or is unknowable, or is a realm of contradictions. See Primacy of Consciousness.
That they make the philosophy dogmatic. The axioms constrain no particular conclusion about anything. What they exclude is a method: appeal to the arbitrary, to faith, or to a claim that reality is other than it is.

Discussion

References

Rand, Ayn. "Axiomatic Concepts", Introduction to Objectivist Epistemology. 2nd edition, Meridian, 1990.
Rand, Ayn. Atlas Shrugged — Galt's speech, on existence, identity and consciousness.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 1.