group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
In the context of étale cohomology of schemes, the Artin comparison theorem (Artin 66) says that under some conditions on a scheme and a sheaf of coefficients, the étale cohomology of the scheme coincides with the ordinary cohomology (e.g. singular cohomology) of its underlying complex analytic topology.
Historically this kind of statement was a central motivation for the development of étale cohomology in the first place.
Specifically, let $A$ be either of
the p-adic integers $\mathbb{Z}_p$ for some prime $p \geq 2$;
the p-adic numbers $\mathbb{A}_{p}$
(but not for instance the integers themselves).
Then for $X$ a variety over the complex numbers and $X^{an}$ its analytification to the topological space of complex points $X(\mathbb{C})$ with its complex analytic topology, then there is an isomorphism
between the étale cohomology of $X$ and the ordinary cohomology of $X^{an}$.
Notice that on the other hand for instance if instead $X = Spec(k)$ is the spectrum of a field, then its étale cohomology coincides with the Galois cohomology of $k$. In this way étale cohomology interpolates between “topological” and “number theoretic” notions of cohomology.
The original reference is
A textbook account is in
Reviews and lecture notes include
James Milne, section 21 of Lectures on Étale Cohomology
In the context of Berkovich analytic spaces see also