History
Abelian categories were introduced by Buchsbaum (1955) (under the name of "exact category") and Grothendieck (1957) in order to unify various cohomology theories. At the time, there was a cohomology theory for sheaves, and a cohomology theory for groups. The two were defined differently, but they had similar properties. In fact, much of category theory was developed as a language to study these similarities. Grothendieck unified the two theories: they both arise as derived functors on abelian categories; the abelian category of sheaves of abelian groups on a topological space, and the abelian category of G-modules for a given group G.
Read more about this topic: Abelian Category