Let A be a subset of group G, then the centralizer of A in G, $C_G(A)$, is not necessarily the center of any group, nor is A necessarily the center of $C_G(A)$. This is true even when A is a subgroup of G. For example, If A = <-i> = {1, -i} subgroup of the quaternion group $Q_8$, then A is not the center of $C_{Q_8}(A)$.
However, the name 'centralizer' makes it sound like $C_G(A)$ should be the center of something. Is this the case?
The centralizer of $a$ is the set of elements that commute with $a$. In other ways, they do the same thing as the elements of the Center do to $a$.