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 Sets. Likewise a C-bundle for a category C is a map whose fibers are functors from C to sets, or, if you want, a disjoint union of sets (one for each object of C) and an action by the morphisms of C - a morphism A-->B in 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.
