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)$. 2 added 14 characters in body I think the answer to your question is Section 29 of Matsumura's book "Commutative ring theory". A summary: A complete$p$-ring is a complete local ring whose maximal ideal is generated by$p$. For any field$k$of characteristic$p$, there is a unique DVR complete$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)$. 1 I think the answer to your question is Section 29 of Matsumura's book "Commutative ring theory". A summary: A complete$p$-ring is a complete local ring whose maximal ideal is generated by$p$. For any field$k$of characteristic$p$, there is a unique DVR whose residue field is$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)\$.