ct.category theory - What does the classifying space of a category classify?

Ieke Moerdijk has written a small Springer Lecture Notes tome addressing this question:"Classifying Spaces and Classifying Topoi" SLNM 1616.

Roughly the answer is: A $G$-bundle is a map whose fibers have a $G$-action, i.e. are $G$-sets (if they are discrete), i.e. they are functors from $G$ seen as a category to $mathsf{Sets}$. Likewise a $mathcal C$-bundle for a category $mathcal C$ is a map whose fibers are functors from $mathcal C$ to $mathsf{Sets}$, or, if you want, a disjoint union of sets (one for each object of $mathcal C$) and an action by the morphisms of $mathcal C$ — a morphism $A to B$ in $mathcal C$ takes elements of the set corresponding to $A$ to elements of the set corresponding to $B$.

There is a completely analogous version for topological categories also.