No two words in your list are equivalent; all of them have their own precise meaning in mathematics:
Axiom (or postulate)
An axiom is a statement that is accepted without proof and regarded as fundamental to a subject. Historically these have been regarded as "self-evident", but more recently they are considered assumptions that characterize the subject of study.
In classical geometry, axioms are general statements while postulates are statements about geometrical objects. A definition is also accepted without proof since it simply gives the meaning of a word or phrase in terms of known concepts.
A theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.
The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules.
The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions.
The concept of a theorem is therefore fundamentally deductive (in contrast to the notion of a scientific theory, which is empirical).
A corollary is a proposition that follows with little or no proof from one other theorem or definition.
A lemma is a "helping theorem", a proposition with little applicability except that it forms part of the proof of a larger theorem. In some cases, as the relative importance of different theorems becomes more clear, what was once considered a lemma is now considered a theorem, though the word "lemma" remains in the name.
Examples include Gauss's lemma and Zorn's lemma.
Conjecture (also: Hypothesis, Claim)
A conjecture is a as-yet unproven proposition that appears correct, for example:
A conjecture becomes a theorem when a formal proof for it becomes established, or until a counter-example or anti-proof determines that it is not true - a good example of this being Fermat's conjecture (often called Fermat's last Theorem for historical reasons), which has now been proven true.
A definition is used to unambiguously define a word for ease of use later (for instance, the definition of a "prime number" being "An irreducible element in the field of Integers").