group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
One says ordinary cohomology for the cohomology theory represented by the Eilenberg-MacLane spectrum for some abelian group .
This is to distinguish from more general notions of cohomology, such as generalized (Eilenberg-Steenrod) cohomology and nonabelian cohomology.
Ordinary cohomology is modeled by singular cohomology.
The refinement to differential cohomology is ordinary differential cohomology.