group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Integral cohomology or “ordinary cohomology” (see there) is the ordinary version of Whitehead-generalized cohomology, the one that is represented by the Eilenberg-MacLane spectrum $H \mathbb{Z}$ with coefficients in the integers.
Integral cohomology is best known maybe in its incarnation as singular cohomology or Čech cohomology with coefficients in the integers.
integral cohomology in degree 1 classifies complex line bundle;
integral cohomology in degree 2 classifies complex line bundle gerbe / line 2-bundles;
integral cohomology in degree $n$ classifies line n-bundles.