classification of $p$-groups

Every $p$-group is a homomorphic image of a $p$-group with cyclic center of order $p$, so a classification (whatever that means) of $p$-groups with cyclic center would (more-or-less) yield a construction for all $p$-groups, and I would not hold my breath waiting for that.

To see why a $p$-group $P$ is a homomorphic image of a $p$-group $G$ with center of order $p$, let $G$ be the regular wreath product of a cyclic group of order $p$ with $P$. Thus $G$ has an elementary abelian subgroup $E$ of order $p^{|P|}$, where $P$ permutes the cyclic factors of $E$ the way it permutes its own elements by right multiplication, and thus $P$ acts faithfully on $E$. Also, $G$ is the semidirect product of $E$ by $P$. It is easy to see that $E \cap {\bf Z}(G)$ has order $p$, so I need to show that every element of ${\bf Z}(G)$ lies in $E$. If $z \in {\bf Z}(G)$, write $z = au$, where $a \in E$ and $u \in P$. Since $z$ centralizes $E$ and $a$ centralizes $E$, it follows that $u$ centralizes $E$ and thus $u = 1$ by the faithfulness of the action. Thus $z = a \in E$, as required.

This was mentioned as a brief comment by Derek Holt, but I think it deserves to be an answer. For a group of order $p^n$ and class $c$, the coclass is $n-c$. The known theory of classification by coclass is much richer than the known theory of classification by class; there is a nice account at http://www.ma.rhul.ac.uk/sepgm/Eick_Classification.pdf. Part of the story is that one can also define coclass for infinite pro-$p$-groups, which is useful for the classification. The isomorphism types of finite $p$-groups of fixed coclass can be assembled into a tree in a natural way, and the infinite paths from the root of the tree correspond to isomorphism classes of pro-$p$-groups.

To echo Marty's sentiment, a classification of $p$-groups based on nilpotency class and size seems far off. As an example of where this problem can get complicated, see

http://www.degruyter.com/view/j/jgth.2011.14.issue-6/jgt.2010.081/jgt.2010.081.xml.

Basically, Halasi and Palfy construct a collection of $p$-groups (allowing $p$ to vary) of nilpotency class 2 where the number of conjugacy classes is not polynomial in the prime $p$. This is related to the (unsolved) question of whether the number of conjugacy classes of $UT_n(\mathbb{F}_q)$, the group of unipotent upper-triangular matrices over the field with $q$ elements, is polynomial in $q$.

To summarize, $p$-groups, even those of nilpotency class 2, get complicated.