de Rham cohomology and flat vector bundles

As you can see from the quick and varied response (is 5 answers in 40 minutes some kind of record?!) the construction is both very useful and has many applications.

One more topic to add to the list, which ties in very nicely with David Speyer's answer, is the link between so-called Higgs fields and flat bundles over Kahler manifolds. This theory, originally due to Hitchin in 1987, is now very much back in vogue because of the role it plays in geometric Langlands. A good introductory reference is here. I'll give an extremely brief summary too, but the article does a much better job.

Given a holomorphic vector bundle E over a complex manifold, a "Higgs field" is a holomorphic 1-form A with values in End(E) which also satisfies $A\wedge A =0$ (the product combines wedge-product on forms and Lie bracket on endomorphisms). This means that if we add A to the d-bar operator on bundle valued forms we get something with square zero, giving a twisted version of the Dolbeault complex David Speyer mentioned in his answer.

Meanwhile, we can build a Higgs bundle by starting with a flat SL(n,C)-bundle. Choosing a Hermitian metric in the bundle we can split the flat connection into two parts, one unitary the other skew-Hermitian. When the metric satisfies a PDE, called "harmonic", the (0,1)-component of the unitary connection gives a holomorphic structure on the bundle and the (1,0)-component of the skew-Hermitian part gives a Higgs field. A theorem of Donaldson and Corlette tells us we can do this whenever the flat bundle is irreducible (i.e. the corresponding rep of the fundamental group is irreducible). Moreover, this construction gives a 1-1 correspondence between stable Higgs bundles and irreducible flat SL(n,C) bundles.

Given a Higgs bundle arising in this way, we now have two different cohomology groups: the twisted Dolbeault groups of d-bar plus A and the coupled deRham groups of the flat connection. Hodge theory tells us that in fact these groups are equal. This is the starting point for a subject called "non-abelian Hodge theory". It gives, amongst other things, deep restrictions on the fundamental groups of Kahler manifolds.

Warning: The first paragraph of the following is outside my expertise.

I am told this construction is very useful in PDE's. If you have a PDE on some manifold $M$, you can often formulate the vector space of solutions as the kernel of some flat connection on a vector bundle. In particular, I believe that the analytic side of the Atiyah-Singer index theorem is the Euler characteristic of the deRham theory you have described.

I can tell you that the analogous construction is very important in complex algebraic geometry. Given a holomorphic vector bundle on a complex manifold, there is a natural way to define a $d$-bar connection on it. (This mean $\nabla_X$ is only defined when $X$ is a $(0,1)$ vector field.) The cohomology of the resulting deRham-like complex, which is called the Dolbeault complex in this setting, is the same as the cohomology of the sheaf of holomorphic sections of the vector bundle. See Wells' Differential Analysis on Complex Manifolds or the early parts of Voisin's Hodge Theory and Algebraic Geometry.

A flat connection on a vector bundle of dimension determines a local system $\mathcal{G}$ over your base manifold $M$ by parallel transport, i. e. a functor from the fundamental groupoid of $M$ to the category of abelian groups (isomorphic to $\mathbb{R}^n$). Note that, although all fibers are isomorphic to $\mathbb{R}^n$ they are not canonically isomorphic. To a local system one can associate (singular) cohomology groups with local coefficients. There are two basic ways of doing this:

1) Do the same construction as for usual singular homology/cohomology, but take as coefficients of a simplex $f: \Delta^k\to M$ a value in the fiber of $\mathcal{G}$ over $f(barycenter)$. Boundary maps are defined via choosing a path from the barycenter of $\Delta^n$ to the barycenter of a face.

2) For a connected $M$, a local system is essentially equivalent to a representation of $\pi_1(M)$ on $\mathbb{R}^n$. On the singular chains of the universal cover $\widetilde{M}$, there is also a $\pi_1(M)$ action via deck transformations. You can now tensor the singular chain complex of $\widetilde{M}$ with $\mathbb{R}^n$ over the group ring $\mathbb{R}[\pi_1(M)]$ and take then homology/cohomology.

I guess, these cohomology groups with local coefficients should be isomorphic to your ''vector bundle deRham cohomology'', though I know no reference at the moment.

Homology/cohomology groups with local coefficients are good for many things in general. I have seen them mostly in the Serre spectral sequence (computing homology/cohomology groups of fiber bundles) where the $E^2$-term is usually written in terms of them. Another application is Poincare duality for non-orientable manifolds.