As krimson has mentioned, calling a result Euler's theorem somewhat under-specifies it. However, here is a theorem that usually goes by that name. It's worth noting that although this result is simple it is used in an important way in the RSA cryptosystem. We need a definition.

Definition If a,b,n are integers with n>0 we say that a is congruent to b modulo n if n divides a-b.

So for example 4 is congruent to 1 modulo 3. We write phi for the Euler Phi function.

Euler's Theorem If a and n are coprime positive integers then

aphi(n) is congruent to 1 modulo n.

Before we prove this result, note that Fermat's Little Theorem is the special case of this result when n is prime.

Proof: We'll use a little bit of technology to prove the theorem. Consider Zn the ring of integers modulo n. Have a look at that writeup because we'll use some results and notation from it. It is shown there that for an integer a we have a is a unit in Zn if and only if (a,n)=1. So we see from the definition of the Euler Phi function that the multiplicative group of units in the ring Zn has order phi(n). By Lagrange's theorem if we raise an element in a finite group to the power the group order we get the identity element. Thus, if a is as in the statement it satisfies

aphi(n) = 1
The result follows since for two integers x and y we have x=y if and only if x is congruent to y modulo n.