Tor
Tensoring with a module is a friendly operation that nonetheless can destroy injectivity: tensoring an injection need not stay an injection. Tor is the precise record of exactly how much injectivity is lost. Its name comes from torsion, because the very first place this loss shows up — tensoring Z/nZ with another group — produces exactly the torsion phenomena, and Tor neatly captures them in every degree.
Formally, for a right R-module A and a left R-module B, Tor_n^R(A, B) is the n-th left derived functor of the tensor product A ⊗_R -: take a projective resolution P_* -> B, form the complex A ⊗ P_*, and take homology. (One may instead resolve A; the answer is the same.) Then Tor_0(A, B) = A ⊗_R B, and a short exact sequence in either variable yields a long exact sequence of Tor groups, repairing the lost left-exactness of tensoring.
Tor vanishes in all positive degrees exactly when the module being resolved is flat — this is the cleanest characterization of flatness. So Tor measures nonflatness the way Ext measures non-projectivity. Over a PID Tor is concentrated in degrees 0 and 1, and Tor_1 detects the torsion submodule, which is the original source of its name. Tor also appears in the Künneth formula relating the homology of a product to the homologies of its factors.
Over Z, Tor_1(Z/mZ, Z/nZ) ≅ Z/gcd(m,n)Z. Resolving Z/mZ by 0 -> Z -> Z -> Z/mZ -> 0 (first map times m) and tensoring with Z/nZ gives Z/nZ -> Z/nZ (multiplication by m); its kernel is Z/gcd(m,n)Z. So Tor_1(Z/2Z, Z/2Z) = Z/2Z.
Tor_1 of two cyclic groups recovers their gcd.