Homological algebra is, in a very myopic sense, the study of chain and cochain complexes in abelian categories. It is, of course, much more exciting than this, but it all stems from the idea of trying to study (co)chain complexes. For a better introduction to what homological algebra 'really is', the preface of any introductory text on homological algebra should give some idea of what it's all about.

A classic text on homological algebra is Cartan and Eilenberg's book 'Homological Algebra'. A more modern treatment is given in Weibel's 'An Introduction to Homological Algebra'. Almost all algebraic topology books will treat at least some aspects of homological algebra. See also the bibliographies at places such as the wikipedia page, nLab page, and Encyclopedia of Mathematics page.