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Difference between a theorem, a lemma, and a proposition

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Axiom/Postulate

A statement assumed to be true without proof.

Lemma

A minor result whose sole purpose is to help to proof a theorem. It is a stepping stone to reach a theorem.

Theorem

A mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, or book the term theorem is generally reserved for the most important results.

Corollary

A result whose proof relies heavily on a given theorem.

Proposition

A proved and often interesting result, but generally less important than a theorem.

Conjecture

A statement that is unproved, but is believed to be true.

Claim

An assertion that is then proved. It is often used like an informal lemma.

Paradox

A statement that can be shown to be both true and false. Paradoxes are often used to show the inconsistencies in a flawed theory .