Just in case you've never heard of (co)homology of arithmetic groups, I've written a quick introduction to why in the world anyone would ever care about this. Hopefully it's not all too dry.
Some Background
Number theorists care about objects known as number fields. These are just finite field extensions of the field Q of rational numbers, such as the field of rational complex numbers Q(i), or the cyclotomic fields Q(ζ). If you've ever taken a class on Galois theory, it should come as no surprise that such number fields correspond to certain subgroups of this large group called the absolute Galois group of Q, Gal(Q), which is just some big group of functions. Because of this, number theorists care about the structure of this group.
As with most groups, one of the most fruitful ways to study Gal(Q) is through its representations, but in contrast to the representation theory of finite groups, these are very difficult to construct in general. However, the Langlands programme can help us here. It explicitly allows us to construct Galois representations from certain functions called automorphic forms, and in the case of GL(2, R) this is the classical theory of modular forms and their associated 2-dimensional Galois representations.
Such modular forms can be seen (via the Eichler-Shimura isomorphism) as certain elements of the cohomology groups of certain spaces (called modular curves). Such cohomology groups identify with the cohomology H^1(Γ, C) of associated groups, called arithmetic groups Γ (or more precisely, arithmetic subgroups of GL(2, R)), so we can view the Langlands programme in this case as associating a 2-dimensional Galois representation to each element of the cohomology H^1(Γ, C) of these arithmetic groups.
We can do this not just for different groups, but for different number fields instead of Q. For example, the Langlands programme for GL(2) over a real quadratic field Q(√d) gives us the theory of Hilbert modular forms, for GL(2) over an imaginary quadratic field Q(√-d) we recover the theory of Bianchi modular forms, and for Sp(2n) over Q we get the theory of Siegel modular forms.
Integral Coefficients and Torsion
The choice of complex coefficients arises naturally in the context of modular forms, but in the more general formulation of the cohomology of arithmetic groups it is an odd one. One might then ask what happens when we change coefficients, say to Z. Since in Z almost nothing is invertible, the change to integral coefficients introduces some torsion into our groups, or rather it no longer eliminates this torsion as C does. So in order to construct a similar Langlands correspondence, we need to see how we can associate Galois representations to these new torsion classes.
One way to construct interesting 2-dimensional Galois representations is to combine two Galois characters (1-dimensional Galois representations), which are governed by class field theory and thus comparatively easy to construct. Such combinations are called extensions, and they are governed by certain Galois cohomology groups, so to construct a non-trivial extension it suffices to construct a non-trivial element of these cohomology groups. So one of the hopes for this theory is to construct an interesting map from certain torsion classes in H*(Γ, Z) to these Galois cohomology groups.
Some Interesting Reads
For the Langlands programme in specific cases
D. Harari, Galois Cohomology and Class Field Theory (CFT is just 1-dimensional Langlands)
H. Jacquet and R. Langlands, Automorphic forms on GL(2)
For the cohomology of arithmetic groups
G. Harder, Cohomology of Arithmetic Groups
F. Calegari and A. Venkatesh, A Torsion Jacquet-Langlands Correspondence