In a group (a special set with an operation on it like addition or multiplication), elements have orders. Usually, on these pages, the group is the set of non-zero remainders modulo a prime and the order of a modulo p then is the least positive integer n such that an 1 (mod p).
For example, let us use a=3 and p=7. Look at the powers of 3 modulo 7:
31 3, 32 2, 33 6, 34 4, 35 5, 36 1The order of 3 modulo 7 is 6. The order of 2 modulo 7 is 3. The order of 6 modulo 7 is 2.
When working modulo a prime, the set of non-zero remainders form a multiplicative group. This is not true modulo a composite. Fermat's Theorem tells us the order of a non-zero element modulo a prime divides the prime minus one. Euler's theorem gives us a similar result for composites.