Conjugacy is one of the more important concepts in group theory.
It's an equivalence relation, an easy exercise.
The class equation gives, for a finite group, the sizes of the various conjugacy classes. This equation is very useful. For instance you can read off the size of the center (the sum of the ones, since elements of the center are in classes of size one), and often tell which subgroups are normal (unions of conjugacy classes).
For an abelian group we just get a string of ones.
Functions that are constant on conjugacy classes are called class functions. As an example there's characters. They were important in Feit-Thompson's theorem, leading to the classification of finite simple groups.
One of the most basic types of group action is conjugation by an element of the group.
The notion also comes up often when dealing with free groups.
Finally for now, conjugate elements in a group always have the same order (since conjugation by an element of the group is an automorphism). This comes in handy when working with group presentations, among other things. For instance when two elements are conjugate and have relatively prime order, they are the identity element.