first cohomology
The first cohomology group answers: given a group acting on something, in how many essentially different ways can you twist a fixed point so that it is no longer fixed but is still compatible with the action? When the action is trivial these twists are just homomorphisms; when the action is nontrivial they are subtler twisted homomorphisms. H^1 organizes all of them, after discarding the twists that come for free from the ambient module.
For a group G and a G-module A, the first cohomology group is H^1(G, A) = Z^1(G, A) / B^1(G, A). Here Z^1 consists of crossed homomorphisms (1-cocycles): functions f : G -> A with f(gh) = f(g) + g·f(h). The subgroup B^1 consists of principal crossed homomorphisms: those of the form f(g) = g·a - a for some a in A. So H^1(G, A) is crossed homomorphisms modulo principal ones. If G acts trivially on A this reduces to H^1(G, A) = Hom(G, A).
Geometrically and arithmetically H^1 classifies objects: for nonabelian A one defines a pointed set H^1(G, A) classifying principal homogeneous spaces (torsors) under A, which is the engine behind descent and the classification of twisted forms. A central tool is the long exact sequence: a short exact sequence of G-modules 0 -> A -> B -> C -> 0 yields ... -> B^G -> C^G -> H^1(G, A) -> H^1(G, B) -> ..., turning the failure of fixed points to surject into a concrete H^1 class.
For G = Z/2Z = {1, σ} acting on A = Z by σ·a = -a, a crossed homomorphism is determined by f(σ) = m with the constraint f(σ^2) = f(1) = 0 forcing m + σ·m = m - m = 0 automatically; coboundaries are f(σ) = σ·a - a = -2a. So H^1(Z/2Z, Z) = Z / 2Z = Z/2Z.
A small H^1 computed directly from cocycles modulo coboundaries.
Hilbert's Theorem 90 is the statement H^1(G, L^*) = 0 for the Galois group G of a finite extension L/K acting on the multiplicative group L^*. This vanishing of a first cohomology group is one of the most consequential computations in all of algebra.