There are two great first examples of complete discrete valuation ring with residue field $\mathbb{F}_p = \mathbb{Z}/p$:  The $p$-adic integers $\mathbb{Z}_p$, and the ring of formal power series $(\mathbb{Z}/p)[[x]]$.  Any complete DVR over $\mathbb{Z}/p$ is a ring structure on left-infinite strings of digits in $\mathbb{Z}/p$.  The difference between these two examples is that in the $p$-adic integers, you add the strings of digits with carries.  (In any such ring, you can say that a $1$ in the $j$th place times a 1 in the $k$th place is a 1 in the $(j+k)$th place.)  At one point I realized that these two examples are not everything:  You can also add with carries but move the carry $k$ places to the left instead of one place to the left.  The ring that you get can be described as $\mathbb{Z}_p[p^{1/k}]$, or as the $x$-adic completion of $\mathbb{Z}[x]/(x^k - p)$.  This sequence has the interesting feature that the terms are made from $\mathbb{Z}_p$ and have characteristic $0$, but the ring structure converges topologically to $(\mathbb{Z}/p)[[x]]$, which has characteristic $p$.

(**Edit:** Per Mariano's answer, I have in mind a discrete valuation in the old-fashioned sense of taking values in $\mathbb{Z}$, not $\mathbb{Z}^n$.)

I learned from Jonathan Wise in [a question on mathoverflow][1] that these examples are still not everything.  If $p$ is odd and $\lambda$ is a non-quadratic residue, then the $x$-adic completion of $\mathbb{Z}[x]/(x^2-\lambda p)$ is a different example.  You can also call it $\mathbb{Z}_p[\sqrt{\lambda p}]$.

So my question is, is there is a classification or a reasonable moduli space of complete DVRs with residue field $\mathbb{Z}/p$?  Or whose residue field is any given finite field? Or if not a classification, an indexed family that includes every example at least once?  

I suppose that the question must be related to the Galois theory of $\mathbb{Q}_p$; maybe the best answer would be a relevant sketch of that theory.  But part of my interest is in continuous families of DVR structures on the Cantor set of strings of digits in base $p$.

  [1]: http://mathoverflow.net/questions/7840/