As far as I understand Deligne's far reaching generalisation of Čech cohomology is called cohomological descent and is used to endow *any* variety with a (mixed) Hodge structure.
Again, AFAIU, the idea is to resolve the variety, then take intersections of the exceptional pieces, resolve those and so on.


As you can see I don't understand it well, so can someone please help? Also, what is the most ridiculously easy example one can have? (I guess the thing I'm interested in most is very simple examples illustrating the possible behaviours)


The only places I know that discuss this are an SGA, Brian Conrad's notes and Peters-Steenbrink.



what is Deligne's cohomological descent (and what are some examples)
As you said, it is a generalization of Cech theory. The standard example to understand first is a divisor $D=\bigcup D\_i$ with simple normal crossings. A resolution of singularities is obtained by simply taking a disjoint union of components $X\_0= \coprod D\_i$. Let $\pi\_0:X\_0\to D$ be the obvious map. The cohomology of $X\_0$ with your favourite coefficients will not (usually) be the same as $D$. So you want to correct this by adding in "higher simplicies". Let $X\_1$ be the disjoint union $\coprod D\_i\cap D\_j$. This has a pair of "face" maps to $X\_1\to X\_0$. We can continue this process to get a (strict) simplicial object
$$ \ldots X\_1\rightrightarrows X\_0\to D$$
with an augmentation to $D$. Given a sheaf $F$ on $D$, we can pull it back to a get a collection of sheaves $F\_i$ on $X\_i$ with various structure maps. We can define the cohomolgy
$H^i(X\_\bullet, F\_\bullet)$ by taking (for example) by compatible injective resolutions, applying $\Gamma$, and taking cohomology of the total complex of the resulting double complex. Now the point is that the machinery of descent tells you that
$$H^i(D, F)\cong H^i(X\_\bullet, F\_\bullet)$$
or if you prefer, there is a spectral sequence relating $H^\*(X\_p, F\_p)$ and $H^\*(D,F)$.
This reduces down to a Mayer-Vietoris type sequence 
$$\ldots H^i(D,F)\to H^i(D\_1, F)\oplus H^i(D\_2,F)\to H^i(D\_1\cap D\_2, F)\ldots$$
when $D$ has two components. Why is this good? Because for certain things, e.g.constructing mixed Hodge structures, it's better to replace the singular space by a bunch of smooth spaces.




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As per request, I'm expanding my comment, although this may be a bit too concise. If $X\_\bullet\to X$ and $Y\_\bullet\to X$ are two
simplicial resolutions, take the fibre product to get simplicial scheme 
$Z\_n =\coprod\_{i+j=n}X\_i\times\_X Y\_j$ dominating both. However, it will be singular. You can build simplicial resolution $\tilde Z\_\bullet$ of this inductively.
First resolve singularites of $Z\_0$ to get $\tilde Z\_0$. For the next step, choose a resolution of $Z\_1$ which maps to $\tilde Z\_0\times\_{X}\tilde Z\_0$ etc.


