Definition: Theorem, Lemma, Proposition, Conjecture and Principle etc.

I have taken this excerpt out from How to think like a Mathematician

• Definition: an explanation of the mathematical meaning of a word.
• Theorem: a very important true statement that is provable in terms of definitions and axioms.
• Proposition: a statement of fact that is true and interesting in a given context.
• Lemma: a true statement used in proving other true statements.
• Corollary: a true statement that is a simple deduction from a theorem or proposition.
• Proof: the explanation of why a statement is true.
• Conjecture: a statement believed to be true, but for which we have no proof.
• Axiom: a basic assumption about a mathematical situation (model) which requires no proof.

I think it does a great job of describing what those words mean in a sentence. Later in the chapter, he goes onto describe how we have some conjectures which have been called "Theorems" even though they weren't proven. For example, Fermat's Last Theorem was referred to as a Theorem even though it hadn't been proven. If you haven't read the book then I highly recommend it if you are a undergraduate in your first two years of math.