See old blog post on this.
Presumably this is for smooth projective varieties over a field.
For a list of cohomology theories related to smooth projective varieties/sheaf cohomology/pure motives, see label: Pure
An attempt to organise the CTs:
We consider smooth projective varieties over a field . See notes from André, under Algebraic cycles and Pure motives, for an introduction to the idea of pure motives. Particular instances of pure motives include Classical motives, Chow motives, Grothendieck motives, Artin motives. In relation to the structure of a general cat of pure motives, one studies Finite-dimensional motives (Kimura, O’Sullivan).
The universal property of pure (Chow) motives is intimately related to the axioms for a Weil cohomology theory. Examples of Weil cohomology theories include: Betti cohomology, Algebraic de Rham cohomology, Crystalline cohomology, l-adic cohomology, Etale cohomology. A Lefschetz operator on a Weil cohomology induces a decomposition; see Primitive cohomology.
A key object of study in all this is the notion of Algebraic cycles, for example Chow groups, also called Chow homology. On of the main motivations for studying cohomology is the desire to understand algebraic cycles.
A key tool for constructing Weil cohomologies is Sheaf cohomology, like étale and crystalline above. Here we should also mention Zariski cohomology, Flat cohomology, and Cech cohomology, although these are not directly related to Weil cohomologies.
Am not sure about how ordinary de Rham cohomology fits in. Same for Motives for absolute Hodge cycles
In order to understand the enriched structure of target cats for Weil cohomology, it would be very interesting to understand the idea of Homotopy type, as described by Toen.
It might be interesting to explore the ideas involved in non-standard approaches to cohomology in algebraic geometry.
nLab page on D30 More about Weil cohomology theories