So only recently encountering conjugation (in the group-theory sense) in my math adventures/education, and I can't help but ask why? It doesn't seem (at first glance) why its worthwhile defining such a term/homomorphism/idea. What do they really tell us about group structure? In $S_n$ they have the nice interpretation of equivalent cycle structures. For finite groups, conjugates can be thought of as having the same cycle structure in the encompassing symmetric group. But since a group on $n$ elements has far less than $n!$ elements, this interpretation isn't so useful.
Can someone offer an interpretation of what these equivalence classes are in a general group? Is the only reason to define them as such is so that we can define quotient groups?
Thanks for any help. Sorry if the post is broad/verbose/may not have an answer.