group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $A$ an abelian Lie group (often taken to be the circle group $U(1)$), a bundle gerbe on $X$ is a representation of a cocycle $c$ in $\mathbf{H}(X,\mathbf{B}^2 A)$.
If a central extension $A \to \hat G \to G$ is given (often taken to be $U(1) \to U(n) \to P U (n)$) there is a notion of $\hat G$-twisted bundles with twist given by $c$.
A bundle gerbe module is the presentation of such a $\hat G$-twisted bundle corresponding to the presentation of the $\mathbf{B}^2 A$-cocycle by a bundle gerbe.
If $Y \to X$ is the surjective submersion relative to which the bundle gerbe $c$ is defined, and if
is the transition line bundle of the bundle gerbe, then a bundle gerbe module for $c$ is a Hermitean vector bundle
equipped with an action
(where $\pi_1, \pi_2 : Y \times_X Y \to Y$ are the two projections out of the fiber product)
that respects the bundle gerbe product
in the obvious way.
When $Y = \coprod_i U_i$ comes form an an open cover $\{U_i \to X\}$ the above almost manifestly reproduces the explicit description of twisted bundles given there.
The concept of bundle gerbe modules was introduced in
for modelling twisted K-theory by twisted bundles.
A splitting principle for bundle gerbe modules is discussed in
For more see at twisted vector bundle.