How is the classifying space BG related to Spec(k[G]) for some ring k, for an Abelian group G?

Question

Given an Abelian group G, we can construct two kinds of “spaces”:

1. BG: a homotopy type, an Elilenberg-Maclane space, the base of a universal G-bundle, e.t.c.
2. Spec(Z[G]): the spectrum of the group algebra over integers Z.

Motivation: I am actually interested in the non Abelian G case. I have heard the G-modules can be thought of as sheaf of modules on BG. I am wondering how that works.