exact category
To build K-theory you need to count objects with the rule that a middle object equals the sum of its ends in any short exact sequence. But many natural collections — finitely generated projective modules, vector bundles, filtered objects — are not abelian categories, so they have no intrinsic notion of exactness. An exact category is the minimal extra structure that fixes this: you simply declare which short sequences count as exact, subject to sensible axioms, and that is enough to run the whole K-theory machine.
An exact category, in Quillen's sense, is an additive category A together with a chosen class of sequences 0 -> A -> B -> C -> 0 called the admissible short exact sequences (or conflations), where the maps A -> B are admissible monics and B -> C admissible epics. The chosen class must satisfy axioms: identities are admissible, the class is closed under isomorphism and contains split sequences, admissible monics and epics are stable under pushout and pullback respectively, and admissible monics/epics compose. Equivalently, A is a full additive subcategory of an abelian category, closed under extensions, with the exact sequences inherited from the ambient abelian category.
Given an exact category A, Quillen's Q-construction produces a topological space whose homotopy groups are the K-groups K_n(A); in degree zero this recovers the Grothendieck group K_0(A) generated by objects modulo [B] = [A] + [C] for each admissible exact sequence. This is the framework in which K-theory of schemes (via vector bundles or coherent sheaves) and of rings is defined uniformly, and in which the localization and resolution theorems are proved.
The category of finitely generated projective R-modules is exact, with admissible exact sequences the ones that split (every short exact sequence ending in a projective splits). Its K-theory is the K-theory of the ring R. By contrast all finitely generated R-modules form an abelian category, whose K-theory is the G-theory of R.
Projectives form an exact category; its K-theory is K(R).
Every abelian category is exact with all its short exact sequences chosen, and every full extension-closed subcategory inherits an exact structure. The slogan is that exactness is structure you choose, not a property you have — which is what lets K-theory apply to categories that are not abelian, such as finitely generated projectives.