What is classified by cohomology?

Hi all categorists Here is another questions i think about and need your help with. 3> What does the cohomology H^n(X;coefficients) classify, for X a more general object then a group and especially when X is a category? I know that the case X=group gives n-torsors. And how come that the classification is independant of the coefficients? Best regards Rafael Borowiecki