group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Similar to the discussion at cohomology group, a cohomology set $H^\bullet(X,A)$ may have the structure of a ring if the coefficient object $A$ suitably does.
In particular for $\mathbf{H} = L_{whe}$Top, then $H^*(X,A)$ is a ring when $A$ is an $E_\infty$-ring.