fiber sequence/long sequence in cohomology
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
By a cohomology theory is usually meant more precisely an example of generalized (Eilenberg-Steenrod) cohomology regarded as a functor that assigns abelian groups or even rings to topological spaces, subject to some conditions. Note that the subject of cohomology theories is itself simply called cohomology.
Mike Hopkins, Complex oriented cohomology theories and the language of stacks course notes (pdf)
Jacob Lurie, A Survey of Elliptic Cohomology - cohomology theories