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An axiom is an irreducible truth expressed in the form of a proposition. A truth you cannot prove nor do without. Axioms are self-evident concepts that form the basis for all higher-level knowledge. Ayn Rand identified three axioms that meet these criteria:

Axioms and Axiomatic Concepts

In her Introduction to Objectivist Epistemology, she describes the nature of axioms, and shows that the most basic units of man's knowledge are not propositions, but Axiomatic Concepts ("Existence", "Identity", "Consciousness"), which she described as follows:

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. It is implicit in all facts and in all knowledge. It is the fundamentally given and directly perceived or experienced, which requires no proof or explanation, but on which all proofs and explanations rest.

Ayn Rand, Introduction to Objectivist Epistemology, Meridian, New York City, 1990, p. 55.

A more lengthy discussion of axioms and their role can be found in OPAR Leonard Peikoff Objectivism: The Philosophy of Ayn Rand.

Axioms are perceptually self-evident, primary, fundamental, and irrefutable. For example, the axiom of identity states that existence is identity, or that to be, is to be something. This axiom is perceptually self-evident because identity is affirmed in everything we experience. Every perception is of something -- having certain properties and not others. Identity is primary because it cannot be broken down into any lower concepts or proven using any lower-level facts. One can only prove that it is an axiom, not the axiom itself, because the concept of proof depends on the validity of identity. It is fundamental because all our higher-level knowledge rests on the fact that existence is identity. If we weren't sure of the axiom, we couldn't know whether contradictory ideas could be true, or whether facts true today would hold true tomorrow. It is irrefutable because any attempt to reject identity must rely on identity being true. For example, if one argues that “facts can be both true and false at the same time”, one assumes that the idea of facts being both true and false is itself true and not false.

Further Reading

Metaphysics Topics
Axioms: Existence, Consciousness, IdentityCorollary: Causality, Primacy of Existence
Reality: Primacy of Existence, Existents, the Metaphysically Given versus the Man Made