I will be so thankful if someone help me about the following question. I need to know the presentation of a (if it is possible) family of finite nonabelian $p$group $G$ with the follwing properties: 1 all noncentral element have abelian centralizer. 2 $cs(G)$ has at least three integer, where by $cs(G)$ I mean the set of all conjugacy class sizes of $G$.

The dihedral groups of order $2^n$ (with $n \geq 4$) form such a family. Indeed, for such a group, we have $$\operatorname{cs}(G) = \{ 1, 2, 2^{n2} \} , $$ and they do have the required property that each noncentral element has an abelian centralizer. Added. Here is another class of examples for arbitrary $p$, still with $\lvert\operatorname{cs}(G)\rvert = 3$ however. Let $N$ be an arbitrary abelian $p$group admitting a nontrivial action of $C_p$ (the cyclic group of order $p$), and let $G$ be the semidirect product $$ G = N \rtimes C_p .$$ Then I claim that all noncentral elements of $G$ have abelian centralizer, and that $$\operatorname{cs}(G) = \{ 1, p, [N:Z(G)] \} . $$ There are three types of elements:
An example of such groups is the wreath product $C_{p^n} \wr C_p$, but of course there are many more examples of this type. 


Consider $p$groups of maximal class with abelian subgroup of index $p$, and order at least $p^4$ (to satisfy second condition in question) $C_p\wr C_p$ is one such group, but order of this (these) group(s) is(are) bounded by $p$, whereas, $p$groups of maximal class, of order $p^n$, with abelian subgroup of index $p$, exists for all $p$ and all $n\geq 3$; the book of LeedhamGreen and S. McKay has an interesting example (see link, Ex. 3.1.5, p. 53) 

