Kisin's work is fairly technical, and is devoted to studying deformations of Galois representations which arise by taking $\overline{K}$-valued points of a finite flat group
over $\mathcal O_K$ (where $K$ is a finite extension of $\mathbb Q_p$).
The subtlety of this concept is that when $K$ is ramified over $\mathbb Q_p$ (more precisely,
when $e \geq p-1$, where $e$ is the ramification degree of $K$ over $\mathbb Q_p$), there
can be more than one finite flat group scheme modelling a given Galois represenation.
E.g. if $p = 2$ and $K = {\mathbb Q}\_2$ (so that $e = 1 = 2 - 1$), the trivial character
with values in the finite field $\mathbb F_2$ has two finite flat models over $\mathbb Z_2$;
the constant etale group scheme $\mathbb Z/2 \mathbb Z$, and the group scheme $\mu_2$
of 2nd roots of unity.
In general, as $e$ increases, there are more and more possible models. Kisin's work shows that they are in fact classified by a certain moduli space (the "moduli of finite flat group schemes" of the title). He is able to get some control over these moduli spaces, and hence prove new modularity lifting theorems; in particular, with this (and several other fantastic ideas) he is able to extend the Taylor--Wiles modularity lifting theorem to the context of arbitrary ramification at $p$, provided one restricts to a finite flat deformation problem.
This result plays a key role in the proof of Serre's conjecture by Khare, Wintenberger, and Kisin.
The detailed geometry of the moduli spaces is controlled by some Grassmanian--type structures that are very similar to ones arising in the study of local models of Shimura varieties. However, there is not an immediately direct connection between the two situations.
EDIT: It might be worth remarking that, in the study of modularity of elliptic curves,
the fact that the modular forms classifying elliptic curves over $\mathbb Q$ are themselves
functions on the moduli space of elliptic curves is something of a coincidence.
One can already see this from the fact that lots of the other objects over $\mathbb Q$ that are not elliptic curves are also classified by modular forms, e.g. any abelian variety
of $GL_2$-type.
When one studies more general instances of the Langlands correspondence, it becomes increasingly clear that these two roles of elliptic curves (providing the moduli space,
and then being classified by modular forms which are functions on the moduli space) are independent of one another.
Of course, historically, it helped a lot that the same theory that was developed to study the Diophantine properties of elliptic curves was also available to study the Diophantine properties of the moduli spaces (which again turn out to be curves, though typically not elliptic curves) and their Jacobians
(which are abelian varieties, and so can be studied by suitable generalizations of many of the tools developed in the study of elliptic curves). But this is a historical relationship between the two roles that elliptic curves play, not a mathematical one.
For the groups $\Gamma$ in Conway-Norton, there is always a moduli problem of $\Gamma$-structures, but since the groups always contain $\Gamma_0(N)$ for some $N$, you won't be able to construct a universal family (because there is a $-1$ automorphism in the way). However, you will sometimes get a ``relatively representable'' problem in the sense of Katz-Mazur.
The upper half-plane quotients will be coarse spaces parametrizing objects of the following general form: You have a diagram of elliptic curves, with some isogenies of specified degrees between them, together with some data that tell you how much symmetry in the diagram you should remember. Since all of the groups normalize $\Gamma_0(N)$ for some $N$, the diagrams will typically involve cyclic isogenies of degree $N$ in some way, and the symmetrization will involve a subgroup of the finite quotient $N_{SL_2(\mathbb{R})}(\Gamma_0(N))/\Gamma_0(N)$.
The standard example is $\Gamma_0(p)^+$ for a prime $p$, which is generated by $\Gamma_0(p)$ as an index two subgroup, together with the Fricke involution $\tau \mapsto \frac{-1}{p\tau}$. The $\Gamma_0(p)$ quotient parametrizes diagrams $E \to E'$ of elliptic curves equipped with a degree $p$ isogeny between them. Taking the quotient of the moduli problem by the Fricke involution amounts to symmetrizing the diagram, so the $\Gamma_0(p)^+$ quotient parametrizes tuples $( \{ E_1, E_2 \}, E_1 \leftrightarrows E_2)$ of unordered pairs of elliptic curves, with dual degree $p$ isogenies between them. Equivalently, you can ask for a set of diagrams $\{E_1 \to E_2, E_2 \to E_1 \}$ where the maps are dual isogenies.
A less well-known example is the 3C group, which is an index 3 subgroup of $\Gamma_0(3|3)$, with Hauptmodul $\sqrt[3]{j(3\tau)} = q^{-1} + 248q^2 + 4124q^5 + \dots$. This group is labeled $\Gamma_0(3|3)$ in the Conway-Norton paper, because $\Gamma_0(3|3)$ is the eigengroup, namely the group that takes the Hauptmodul to constant multiples of itself. The 3C group contains $\Gamma_0(9)$ as a normal subgroup, with quotient $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. You can view the upper half-plane quotient as a parameter space of quadruples of elliptic curves, with a rather complicated system of cyclic 9-isogenies and correspondences that get symmetrized (more on this in the last paragraph). A more succinct expression follows from using the matrix $\binom{30}{01}$ to conjugate $\Gamma_0(3|3)$ to $\Gamma(1)$ and $\Gamma_0(9)$ to $\pm \Gamma(3)$. Then you're basically looking at a moduli problem that parametrizes elliptic curves $E$ equipped with an unordered octuple of symplectic isomorphisms $E[3] \cong (\mathbb{Z}/3\mathbb{Z})^2$ that form a torsor under the characteristic 2-Sylow subgroup $Q_8 \subset Sp_2(\mathbb{F}_3) \cong SL_2(\mathbb{Z})/\Gamma(3)$.
In general, you can encode moduli problems attached to arithmetic groups using the fact that congruence groups like $\Gamma(N)$ and $\Gamma_0(N)$ stabilize distinguished finite subcomplexes of the product of all $p$-adic Bruhat-Tits trees. Conway gives a explanation (that doesn't use the word "moduli") with pictures in his paper Understanding groups like $\Gamma_0(N)$. For example, when $N$ is a product of $k$ distinct primes, $\Gamma_0(N)$ stabilizes a $k$-cube. Given a finite stable subcomplex, there is a standard way to make a moduli problem out of it by assigning elliptic curves to the vertices, isogenies to the edges, such that the induced transformations on the Tate module behave as you would expect from traversing the product of buildings. To symmetrize, just enumerate orbits of the transformations you want, and demand a torsor structure.
In the case of the 3C group in the above paragraph, $\Gamma_0(9)$ pointwise stabilizes a subgraph of the 3-adic tree that is an X-shaped configuration spanned by 5 vertices. The edges coming out of the central vertex are in noncanonical bijection with points in $\mathbb{P}^1(\mathbb{F}_3)$, and to symmetrize, you can make an unordered 4-tuple of diagrams of 5 elliptic curves, related by the action of the subgroup $V_4 \subset PSL_2(\mathbb{F}_3)$ that preserves the cross-ratio.
Best Answer
The first is unrepresentable for arbitrary large $N$ (it depends on the residue class of $N$ mod 12), the second is representatble for $N \geq 4$ (if you are considering $Y_1(N)$) or $N \geq 5$ (if you are considering $X_1(N)$, i.e. including the cusps), the third is representable for $N \geq 3$.
The references you mentioned are the standard ones. Probably Silverman discusses these in his books somewhere too (maybe the 2nd). If you look in Gross's Duke paper on companion forms (A tameness criterion ... ) you will find a summary of the story for $X_1(N)$. In the $\Gamma_0(N)$ case, Mazur has a careful discussion in the beginning of section 2 of his Eisenstein ideal paper. Both Gross and Mazur refer back to Deligne--Rapoport for proofs.
It is also just a matter of computing the torsion in each of the $\Gamma$'s (plus epsilon more if you want to understand representability at the cusps), which is an exercise. (Although you have to do a little work to see why this is the necessary computation.)