An axiom is an assumed property of a formal system. Generally, this means that it can't be proved that the axiom is true. Rather, it is assumed to be true, and may even be self-evident. You can think of it as a "rule of the game". A theorem is a statement within the formal system that can be proved by a series of correct logical deductions arising from axioms (or from theorems that have already been proved).

Number theory (the study of integers) uses a very limited set of axioms (the fewer, the better, obviously). The following list is complete to the best of my knowledge and belief:

1. For all m and n, it is true that m + n = n + m (commutative law of addition), and that mn = nm (commutative law of multiplication).
2. For all m, n, k, it is true that (m + n) + k = m + (n + k) (associative law of addition) and that (mn)k = m(nk) (associative law of multiplication).
3. For all m, n, k, it is true that m(n + k) = (mn) + (mk) (distributive law).
4. There is a number 0, which has the property that for any number n, n + 0 = n (existence of additive identity).
5. There is a number 1, which has the property that for any number n, n × 1 = n (existence of multiplicative identity).
6. For every number n there is another number k such that n + k = 0 (existence of additive inverses).
7. For any m, n, k, if k is not 0 and km = kn, then m = n (cancellation law).

The idea is to build up theorems from axioms. Consider, for example, this proof (which I've pinched from Keith Devlin) that if k + m = k + n, then m = n.

1. Assume that k + m = k + n
2. From Axiom 1, m + k = n + k
3. From Axiom 6, let l be a number such that k + l = 0
4. Adding l to both sides yields: (m + k) + l = (n + k) + l
5. From Axiom 2, m + (k + l) = n + (k + l)
6. Remembering our definition of l is such that k + l = 0, we now have m + 0 = n + 0
7. From Axiom 4, then, we have m = n

That took more work than you might think was strictly necessary, but the point is this: now that we have proved the theorem (that if k + m = k + n then m = n), we don't ever have to prove it again. We can "take it as read", and use it as a fact in other proofs.

All this work has already been done, with the upshot that you can generally be a little lazy with proofs. You don't have to prove everything from axioms. (But you can if you like!)