Conjugate elements (in a group) have the 'same' properties, essentially. This is because the map
f_g :G-->G
f_g(x)=g^{-1}xg
is an isomorphism. The set of all such f_g, g in G is the group of inner automorphisms. In a lot of cases these are all automorphisms of a finite group; in some cases they are not.
In linear algebra, conjugate matrices share many properties....
The number of conjugacy classes of a finite group is the same as the number of simple complex valued representations.
Shall I go on?
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They first seem a bit weird but now that you mentioned these things, they seem quiet natural.