What is the difference between lemma, axiom, definition, corollary, etc?

There is often a difference, although usage might depend on author / language /... I will be as concrete as possible, at the expense of being a bit sloppy:

• Axiom: a fundamental logical statement that you assume to be true in order to build a theory. Nothing grows out of nothing: even to construct logic or mathematics you need to start from some assumptions that you just accept as reasonable.
• Definition: one cannot do mathematics using just logical symbols: it is just too cumbersome. Often one introduces simplifications, notations, names to talk about things that come up frequently. It is an agreement about calling something in a certain way.
• Lemma: a true statement that can be proved (proceeding from other true statements or from the axioms) and that is immediately (or almost immediately) used to prove something more important (a theorem / proposition).
• Theorem: an important and/or difficult to prove true mathematical statement.
• Proposition: a true mathematical statement that is not as important / difficult as a theorem. Let's say, an ordinary true mathematical statement.
• Corollary: a true mathematical statement that follows quite directly as a consequence of a theorem or proposition (e.g. as a special case).
• Law: not very much used in pure mathematics, it is more common e.g. in physics to refer to a true fact about nature.

Note that sometimes tradition gets in the way: you can accept Zorn's Lemma as an axiom, and similarly there are lemmas that have become theorems in theor own right. There is not always a sharp distinction!