[nLab on Classifying topos and Classifying space]
Jardine-Goerss chapter V treats simplicial sets with an action of a simplicial group, classification of principal -fibrations, construction of a model of via universal cocycles.
There are descriptions of the mod p cohomology of the classifying spaces of compact Lie groups, see LNM1370, and Dwyer, Wilkerson: A cohomology decomposition thm (1992).
Dwyer et al: Homotopical uniqueness of classifying spaces
Book: Dwyer, Henn: Homotopy theoretic methods in group cohomology. Explains various things including nerves, equivariant homotopy theory, transfer, the T-functor of Lannes
There are results by Suslin, Jardine, Karoubi and others about bijective maps between cohomology of classifying spaces computed in topological and discrete versions. See http://www.ams.org/mathscinet-getitem?mr=764100 for some discussion about when this holds and a profinite counterexample.
For classifying spaces of algebraic groups in the setting of A1-homotopy theory, see Morel-Voevodsky: A1-homotopy theory of schemes, chapter 4. Notion of etale classifying space, and relation to algebraic K-theory.
http://mathoverflow.net/questions/51694/history-of-classifying-spaces
ALGTOP discussion here, scroll down a bit.
nLab page on Classifying space