Complete discrete valuation rings with residue field ℤ/p

Greg, I want to say some basic things, but people are giving quite "high-brow" answers and what I want to say is a bit too big to fit into a comment. So let me leave an "answer" which is not really an answer but which is basically background on some other answers.

So firstly there is this amazing construction of Witt vectors, which takes as input a finite field $k$ of characteristic p (well, it can take a lot more than that but let me stick with finite fields of characteristic p) and spits out a canonical complete DVR $W(k)$ with residue field $k$ and uniformiser $p$. If you feed in $Z/pZ$ it spits out $Z_p$ and if you feed in, say, the field $Z/5Z[\sqrt{2}]$ with 25 elements it spits out something isomorphic to $Z_5[\sqrt{2}]$, and so on. Witt vectors are just a way of formalising the "carry" business---it gets a bit trickier when the residue field isn't Z/pZ. Turns out that any complete DVR $R$ with fraction field of characteristic 0 and residue field $k$ is canonically and uniquely a $W(k)$-algebra. So that's pretty cool. Furthermore, $R$ will be finite and free over $W(k)$, and $R$ is the "ring of integers" (this makes sense) in its fraction field $K$, which is a finite extension of the fraction field of $W(k)$, something which in turn will be finite over $Q_p$ of degree $d$ if $[k:Z/pZ]=d$. [Note in particular that "fraction field of Witt vectors" gives us a whole bunch of extension of $Q_p$---the so-called "unramified" ones---one for each finite extension of $Z/pZ$.]

Conversely, given an arbitrary finite field extension $K$ of $Q_p$, the ring of integers of $K$ (that is, the elements in $K$ satisfying a monic polynomial with coefficients in $Z_p$) will be a complete DVR with residue field some finite field $k$, and then the field of fractions of $W(k)$ embeds into $K$. This subfield of $K$---the "maximal unramified subextension of $K$" is canonical and intrinsic. An extension $K$ of $Q_p$ is "totally ramified" if the residue field of $R$ is $Z/pZ$. A FABULOUS place to read about this stuff is Serre's book "local fields". Everything there, with complete canonical proofs.

So one aspect of your question is whether there is a moduli space of totally ramified extensions of $Q_p$. I wouldn't rule such a gadget out but I've not seen one. This might be just a question in algebra. If $K$ is a finite totally ramified extension of $Q_p$ then $K=Q_p(\pi)$ with $\pi$ a uniformiser of the integers of $K$, and $\pi$ will satisfy a degree $d$ equation if $d=[K:Q_p]$. Furthermore this degree $d$ equation will be monic, with coefficients in $Z_p$, and the constant term will be $p$ times a unit. Conversely any such equation will give a totally ramified extension of $Q_p$ (they will all be irreducible by Eisenstein's criterion).

So now we have a list of all totally ramified extensions of $Q_p$, and hence all complete DVRs with residue characteristic $Z/pZ$ but generic characteristic zero, because we just list all polynomials of this form. Unfortunately each $R$ is in our list infinitely often. So to make the $R$s the $Z/pZ$-points of a moduli space we need to quotient out the set of degree $d$ Eisenstein polynomials by the relation "the extension of $Q_p$ generated by these polynomials are the same". A map $Q_p(\pi_1)\to Q_p(\pi_2)$ is just another polynomial so it looks to me like there is hope that this can be done, but I've not done it.

Finally, everything I said above has a natural generalisation to any finite field, not just $Z/pZ$.

When people talk about class field theory or Lubin-Tate groups above, what they're saying is that there are certain totally ramified extensions of $Q_p$, namely those which are Galois over $Q_p$ with abelian Galois group, which can be constructed explicitly using other techniques (like formal groups or Artin maps or whatever), and these constructions generalise to give all abelian extensions of an arbitrary finite extension of $Q_p$. However if you're looking for general moduli spaces then it seems to me that these notions might not be of too much use to you because they don't give all extensions, just abelian ones.

There. So really that was just a comment but it was visibly too large. Hopefully someone will now quotient out the Eisenstein polynomials by an equivalence relation thus giving you your moduli space, because that seems to me to be the heart of your question.

The classification of CDVRs with residue field any given perfect field k is discussed in Chapter 2 of Serre's Local Fields. In particular:

Theorem II.2: Let R be a CDVR with residue field k. Suppose R and k have the same characteristic and that k is perfect. Then R is isomorphic to k[[t]].

Theorem II.3: For every perfect field k of characteristic p, there exists a unique CDVR (up to unique isomorphism) which is absolutely unramified [i.e., p is a uniformizing element] and has k as its residue field: namely W(k), the Witt vector ring.

Theorem II.4: Let R be a CDVR of unequal characteristic with perfect residue field k. Let e be its absolute ramification index. Then there exists a unique homomorphism W(k) -> R commuting with reduction modulo the maximal ideal. This is injective, and R is a free W(k)-module of rank e.

Thus the CDVRs with residue field Z/p are: Z/p[[t]] and the valuation ring of a totally ramified extension of Z_p. In particular, the set of such isomorphism classes is countably infinite, and there no moduli in any sense known to me.

I don't know what all the cdvr's with residue field $F_p$ are, but local class field theory gives you a huge family. Those that are finite extensions of $Z_p$ are the rings of integers in the totally ramified extensions of $Q_p$. For each prime element $\pi$ of $Z_p$, there is a unique totally ramified extension $K_{\pi}$ of $Q_p$ such that (a) $\pi$ is a norm from every finite subextension of $K_{\pi}$ and (b) the maximal abelian extension of $Q_p$ is obtained by composing $K_{\pi}$ with the unramified extensions of $Q_{p}$. For example, if $\pi=p$, then $K_{\pi}$ is obtained by adjoining the $p^{n}$th roots of $1$ to $Q_{p}$ for all $n$. In general, $K_{\pi}$ is given quite explicitly by Lubin-Tate theory. Different $\pi$ give different field extensions, so the $K_{\pi}$'s are parametrized by the units in $Z_p$. See, for example, Chapter I of my notes