What exactly does conjugation mean? In group theory, the mathematical definition for "conjugation" is:
$$
(g, h) \mapsto g h g^{-1}
$$
But what exactly does this mean, like in laymans terms?

Conjugation is an important construction in group theory. Conjugation defines a group action of a group on itself and this often yields useful information about the group. For example, this technique is how the Sylow Theorems are proven. More importantly, a normal subgroup of a group is a subgroup which is invariant under conjugation by any element. Normal groups are extremely important because they are the kernels of homomorphisms and it is possible to take the quotient of a group and one of its normal subgroups. 
