group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A Bott element is an element in a periodic cohomology theory, multiplication by which is invertible.
Specifically for complex topological K-theory there is an element of degree 2 which induces Bott periodicity. This is where the term “Bott element” comes from. For details on this case see at fundamental product theorem in topological K-theory.