In mathematics, what is the difference between a theorem and a conjecture?

The difference is in a sense between what we THINK is true and what we KNOW to be true.

A conjecture is a statement that for some reason or other we believe to be true. The reason might be because it's known to hold in some special cases, or because we ran some experiments and it always seemed to hold, or because it just would fit in with other things that we already know to be true.

At some point (hopefully) someone will prove the conjecture (rigorously show that the statement is always true). At that point the conjecture turns into a theorem.

More:

http://www.mathsisgoodforyou.com/conjecturestheorems/conjecturestheorems.htm