I think the answer to your question is Section 29 of Matsumura's book "Commutative ring theory".

A summary: A complete $p$-ring is a complete local ring whose maximal ideal is generated by $p$. For any field $k$ of characteristic $p$, there is a unique complete $p$-ring whose residue field is $k$.
In the case of $k=\mathbb Z/(p)$, that ring is $\mathbb Z_p$. 

We call a DVR $(R,m,k)$ unramified if it $char R= char k$ or if it is mixed characteristic and $p \notin m^2$. We call it ramified if $p\in m^2$. 

Theorem: An unramified, complete DVR is isomorphic to $k[[x]]$ or is the unique $p$-ring whose residue field is $k$. A ramified one is isomorphic to $A[[x]]/(f)$, here $A$ is said unique $p$-ring and $f = p + g(x)$, here $g \in (x^2)$. 

So apart from the 2 unramified ones, the ramified ones are parametrized by the elements
$g \in (x^2)$.