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.

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.