Where ordinary cohomology has as coefficient objects abelian groups, Bredon cohomology has as coefficients functors
O(G) \to Ab
There is an interesting (nontrivially) equivalent definition by Moerdijk and Svensson, using the Grothendieck construction for a certain -valued presheaf on the orbit category.
Further remarks on this are in
G. Mukherjee, N. Pandey, Equivariant cohomology with local coefficients (pdf)
H. Honkasalo, Sheaves on fixed point sets and quivariant cohomology, Math. Scand. 78 (1996), 37–55 (pdf)
H. Honkasalo, A sheaf-theoretic approach to the equivariant Serre spectral sequence, J. Math. Sci. Univ. Tokyo 4 (1997), 53–65 (pdf)
For orbifolds there is a generalization of -theory which is closely related to the Bredon cohomology (rather than usual equivariant cohomology):