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 $char R= char k$ 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 $g \in (x^2)$. n^2)$, with $n=(p,x)$ is the maximal ideal of $A[[x]]$.
So apart from the 2 unramified ones, the ramified ones are parametrized by the elements $g \in (x^2)$. n^2)$.