A category $C$ is homological if it is pointed, regular and protomodular.
Many of the standard results of classical homological algebra in abelian categories extend to homological categories: the five lemma, the nine lemma, the snake lemma, long exact sequence in homology, the Noether isomorphism theorem. There is also a version of the Jordan-Hölder theorem.
A homological category which is Barr-exact and has finite coproducts is semiabelian.
Dominique Bourn, Francis Borceux, Mal'cev, protomodular, homological and semi-abelian categories, Kluwer 2004.
Francis Borceux, Marco Grandis, Jordan-Hölder, modularity and distributivity in non-commutative algebra, J. Pure Appl. Algebra 208 (2007), no. 2, 665–689, doi, MR2007k:18021