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.
In the spirit of first approach, there is a conjecture for a Clifford-algebra type proof of the 576-fold periodicity of TMF. This is a generalized cohomology theory constructed by piecing together all the elliptic cohomology theories together in a suitable way. I heard about this conjecture from Andre Henriques, who is working on a geometric approach to TMF using conformal nets.
The idea is that the free fermion conformal net (a introduction can be found in this article by Bartels, Douglas and Henriques) is to TMF as the Clifford algebras are to K-theory. For a suitable sense of equivalence, i.e. some generalization of Morita equivalence, $Free(n)$ and $Free(n+576)$ should be equivalent. I believe people are still far from a proof, but the motivation comes from looking at orientations: a manifold is orientable for K-theory if the frame bundle (a principal $SO(n)$-bundle) lifts to a principal $Spin(n)$-bundle. The $Spin(n)$ groups can be defined as a group in the Clifford algebra. For TMF, a manifold is orientable if the frame bundle extends to a principal $String(n)$-bundle, which can be obtained from $Spin(n)$ by killing $\pi_3$, just like $Spin(n)$ is obtained from $SO(n)$ by killing $\pi_1$. There is a way to define $String(n)$ using the free fermion conformal nets.
The only reference I know for these ideas is the following summary.
Best Answer
The word modulus (moduli in plural, cf. radius and radii, focus and foci, locus and loci) comes from Latin as a word meaning "small measure" or "unit of measure". This is why the absolute value of a complex number z is sometimes called the modulus of z and why the word is used in physics for Young's modulus. In 1800 Gauss introduced the congruence relation $a \equiv b \bmod m$ with m being called the modulus because this equivalence relation on integers a and b was being "measured" according to the integer m. The later term "modular representation" for representations in characteristic p comes from the simplest source of fields with characteristic p being the integers mod p (and finite extensions of it).
The term modulus, from its meaning as a standard of measure, drifted into a more general usage as "parameter". This meaning led to the terms modular function and modular form by the end of the 19th century.
In the study of elliptic integrals like $u(x) = \int_0^x \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}$ the parameter $k$ that shows up has traditionally been called the elliptic modulus of the elliptic integral. It pins down which elliptic integral you are speaking about, since the integral varies with $k$. This name for k goes back to Jacobi's work on elliptic integrals in the 1820s. Inverting the relation between u and x, $x(u)$ extends from a neighborhood of $u = 0$ to all of $\mathbf C$ as a doubly periodic meromorphic function, and if $\tau$ is a ratio of two carefully chosen periods (taken in the upper half-plane) then Gauss found a formula for k in terms of some theta-functions evaluated at $\tau$. These theta-functions are invariant under a finite-index subgroup of $\text{SL}_2(\mathbf Z)$ (specifically, invariant under $\Gamma(2)$), although such work of Gauss was unpublished until it appeared after his death (1855) in his collected works. Earlier, less than 10 years after Jacobi's work, in volume 18 (1838) of Crelle's Journal, Gudermann wrote a long paper Theorie der Modular-Functionen und der Modular-Integrale (pp. 1-54 and 220-258). Jacobi's modulus k and the Jacobi elliptic functions sn, cn, dn appear prominently in it; while I can't read this German very well, other sources that refer to this paper say Gudermann's use of "modular function" refers to what we would call elliptic functions (and, correspondingly, what he calls "modular integrals" are elliptic integrals). I mention Gudermann's paper just to point out that the term "modular function" was in use before 1840 (hence preceding Riemann's work), even if its meaning over time would change.
Paul Garrett says in an answer to a math stackexchange question here that in the 1870s Dedekind introduced in his work on algebraic number theory the term Modul for what we'd nowadays call a lattice (in Euclidean space) or finite-free $\mathbf Z$-module, and it is suggested in one of the answers here that this term might have been chosen because it was a general kind of structure you could "mod" out by. I think that is what the word eventually came to mean (related to the algebraic notion of a module), but I took a look at Dedekind's famous Chapter XI of Dirichlet and Dedekind's Vorlesungen über Zahlentheorie, where he first introduces the term Modul, in section 168, and he doesn't actually use Modul to refer to lattices at all. Dedekind is thinking of a number field as a subset of $\mathbf C$ (there were no abstract fields in those days, except for perhaps finite fields) and he defines a Modul $\mathfrak a$ as any set (he writes "System") of real and complex numbers closed under subtraction. From any elements $\alpha$ and $\beta$ in $\mathfrak a$ you get $\alpha-\alpha=0$ and then $0 - \alpha = -\alpha$ and then $\beta - (-\alpha) = \alpha+\beta$, so basically a Modul is just an additive subgroup of $\mathbf C$ (not of a general $\mathbf R^n$). Dedekind is most interested in the case when $\mathfrak a$ is a finite Modul, meaning it has a finite basis, such as $\mathbf Z + \mathbf Z\sqrt{2}$ and $\mathbf Z + \mathbf Z\sqrt[3]{2} + \mathbf Z\sqrt[3]{4}$ in $\mathbf R$. Neither of these are lattices, since we are not putting all the real and complex embeddings of a number field together to make such groups discrete in a larger Euclidean space. (In section 177 Dedekind defines an ideal to be a special kind of Modul in a number field.) Perhaps over time the concept of a Modul did turn into a lattice, and at least inside the integers of imaginary quadratic fields, which are so important for complex multiplication, they are the same thing. Lattices in $\mathbf C$, up to real scaling, can be written as $\mathbf Z + \mathbf Z\tau$ with $\tau$ in the upper half-plane being determined by that lattice up to the action of $\text{SL}_2(\mathbf Z)$. This group had been used by Gauss in the early 19th century in his work on binary quadratic forms and the arithmetic-geometric mean, but the study of this group and especially its subgroups really took off in the late 19th century in work of people like Dedekind, Fricke, and Klein. On p. 364 here Klein says the term "elliptic modular function" comes from Dedekind's 1877 article here. What about modular forms? Eisenstein series were introduced in the mid-19th century and they were viewed later as certain homogeneous functions on lattices. The word "form" is a concise term for homogeneous function (like "linear form" and "quadratic form"), so these homogeneous functions on lattices got the name "modular form". According to this article the term "modular form" was introduced by Klein on pp. 143-144 here, and even if you don't know any German you can recognize the German for "homogeneous function" on p. 144. See also pp. 127-128. (From a different direction, in 1909 Dickson wrote a paper where he used "modular form" to mean "homogeneous polynomial over a finite field"!)
Riemann introduced the term "moduli" (Moduln) in his 1857 paper on abelian functions (see bottom of p. 33 here) to refer to a count of $3p-3$ parameters or coordinates, which is the source of the later idea of a moduli space or parameter space for geometric objects of a common type.
In summary, there are several sources of all this "modular" terminology: Gauss's modulus in 1800, Jacobi's elliptic modulus $k$ in the 1820s, Riemann's moduli in the 1850s, and Dedekind's module in the 1870s. I don't see how Riemann's moduli (leading to moduli spaces) is related to the development of the other mod-type terms in the 19th century.