group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
What is called generalized Eilenberg-Steenrod cohomology is really the general fully abelian subcase of cohomology.
This means that generalized Eilenberg-Steenrod cohomology is the cohomology in an (∞,1)-category $\mathbf{H}$ that happens to be a stable (∞,1)-category.
The archetypical example of this is $\mathbf{H} = Sp(Top)$, the stable (∞,1)-category of spectra and this is the context in which generalized Eilenberg-Steenrod cohomology is usually understood. So
Generalized Eilenberg-Steenrod cohomology is cohomology $H(X,A)$ with coefficient object $A$ a spectrum.
One may conceptualize the axioms as ensuring that certain nice properties that hold in the category Top will be preserved by our cohomology functor.
Let $U$ and $X$ be topological spaces, such that $U$ is a subspace of $X$. Notation: $(X,U) := U \hookrightarrow X$.
Note that if only one space is listed, the subspace is assumed to be the empty set $(X, \emptyset)$.
The Eilenberg-Steenrod axioms are the following:
A cohomology theory is a collection $\{A^n\}_{n \in \mathbb{Z}}$ of functors
from the homotopy category of pairs of topological spaces to the category Ab of abelian groups, as well as a natural transformation $\delta: A^n(X, \emptyset) \to A^{n+1}(X, U)$. These functors and natural transformations satisfy and are characterized by the following axioms.
$\cdots \to A^n(X, U) \to A^n(X, \emptyset) \to A^n(U, \emptyset) \xrightarrow{\delta} A^{n+1}(X, U) \to \cdots$
Weak homotopy equivalence: if $f : X \to Y$ is a weak homotopy equivalence then $A^n(f) : A^n(Y) \to A^n(X)$ is an isomorphism
Additivity: If $(X, U) = \coprod_i (X_i, U_i)$, then $A^n(X, U) = \coprod_i A^n(X_i, U_i)$.
Excision: Let $S$ be a subspace of $U$, the natural inclusion of the pair $i:(X-S, U-S) \hookrightarrow (X, U)$ induces an isomorphism $A^n(i): A^n(X-S, U-S) \to A^n(X, U)$.
Ordinary cohomology theories require and additional axiom, the dimension axiom $A^n(pt) = 0$.
twisted generalized cohomology theory is ∞-categorical semantics of linear homotopy type theory:
remark Originally Eilenberg and Steenrod had written down axioms that characterized the behaviour of ordinary integral cohomology, what is now understood to be cohomology with coefficients in the Eilenberg-MacLane spectrum. Generalized Eilenberg-Steenrod cohomology is originally defined as anything that satisfies this list of axioms except the first one. Later it was proven, by the Brown representability theorem, that all the models for these axioms arise in terms of homotopy classes of maps into a spectrum. In our revisionist perspective above, we take this historically secondary point of view as the conceptually primary one.
A pedagogical introduction to spectra and generalized (Eilenberg-Steenrod) cohomology is in
A comprehensive account is in
Generalized cohomology (pdf)
Peter May Concise Course on Algebraic Topology: Chapter 18
More references relating to the nPOV on cohomology include:
Mike Hopkins, Complex oriented cohomology theories and the language of stacks course notes (pdf)
Jacob Lurie, A Survey of Elliptic Cohomology - cohomology theories