I like to say that there is only a single abstract definition of cohomology: in any $(\infty,1)$-topos $\mathbf{H}$ given objects $X$ and $A$, the cohomology of $X$ with coefficients in $A$ is the connected components of the hom-$\infty$-groupoid $H(X,A) := \pi_0 \mathbf{H}(X,A)$.

Everything else one sees described as "cohomology" is, i claim, a special case and a special realization of this situation.

More on this point of view is at <a href="http://ncatlab.org/nlab/show/cohomology">cohomology</a>

In particular, ordinary abelian sheaf cohomology for sheaves on a cite $C$ is the cohomology in this sense of the <a href="http://ncatlab.org/nlab/show/models+for+infinity-stack+(infinity%2C1)-toposes">(oo,1)-topos of oo-stacks on C</a> where the coefficient objects are, moreover, restricted to be objectwise in the image of the <a href="http://ncatlab.org/nlab/show/Dold-Kan+correspondence">Dold-Kan map</a> (are "maximally abelian oo-stacks").

From this perspective the relation betwen Cech-cohomology and other means to compute sheaf-cohomology become conceptually evident: all of these are just models to model the <a href="http://ncatlab.org/nlab/show/(infinity,1)-categorical+hom-space">(oo,1)-cateorical hom-space</a> $\mathbf{H}(X,A)$: Cech cohomology does so by finding cofibrant versions of $X$ (namely <a href="http://ncatlab.org/nlab/show/Cech+nerve">Cech nerves</a> of Cech covers), derived-functor-style sheaf cohomology usually does so by finding fibrant versions of $A$ (namely injective resolutions of sheaves).

That this is the relation between the two is of course implicitly the old Verdier hypercovering theorem. A particularly clear-sighted description of this is the remarkable old article by Kenneth Brown, <a href="http://ncatlab.org/nlab/show/BrownAHT">Abstract homotopy theory and generalized sheaf cohomology</a>.

A summary of that in the light of the above comments is at <a href="http://ncatlab.org/nlab/show/abelian+sheaf+cohomology">nlab:abelian sheaf cohomology</a>.

Technical details are also at <a href="http://ncatlab.org/nlab/show/Cech+cohomology">Cech cohomology</a>.