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.