bar resolution
The bar resolution is the universal recipe that turns the abstract definition of group cohomology into explicit functions you can compute with. Derived functors are defined using any free resolution, but to actually write down cocycles you need a concrete one; the bar resolution supplies the same resolution for every group, built tautologically from the group's own elements. Its name comes from an old notation that separated the elements with vertical bars.
For a group G, the bar resolution is the free Z[G]-resolution of the trivial module Z whose n-th term B_n is the free Z[G]-module on symbols [g_1 | g_2 | ... | g_n] (with B_0 = Z[G] on the empty symbol). The boundary map d : B_n -> B_{n-1} is the alternating sum d[g_1 | ... | g_n] = g_1·[g_2 | ... | g_n] + Σ_{i=1}^{n-1} (-1)^i [g_1 | ... | g_i g_{i+1} | ... | g_n] + (-1)^n [g_1 | ... | g_{n-1}]. Applying Hom_{Z[G]}(-, A) to this resolution yields exactly the inhomogeneous cochain complex of functions G^n -> A.
Thus the bar resolution is why the hands-on formulas for cocycles and coboundaries — the crossed-homomorphism law in degree 1, the factor-set identity in degree 2 — look the way they do: they are simply the duals of the bar boundary maps. The normalized bar resolution, which kills any symbol containing an identity entry, gives a smaller chain-homotopy-equivalent complex of normalized cochains and is what one uses in practice to keep computations finite.
Dualizing the bar resolution in low degrees: Hom(B_1, A) is functions G -> A and Hom(B_2, A) is functions G × G -> A; the coboundary d^1 f(g, h) = g·f(h) - f(gh) + f(g) is exactly the dual of the bar map d : B_2 -> B_1. Its kernel is the crossed homomorphisms.
Dualizing the bar maps recovers the explicit cochain differentials.
The bar resolution is enormous — each B_n is free on |G|^n generators — so it proves theorems but is hopeless for direct computation of large groups. For an actual group one replaces it by a small efficient resolution (e.g. the periodic one for cyclic groups) that computes the same cohomology.