An introduction to homological algebra, by C. Weibel, Cambridge U. Press
First-year knowledge of groups and modules.
From some perspectives, homological algebra is the study of the failure of modules over rings (and related objects) to behave like vector spaces. Somewhat more precisely, homological algebra collects a number of ideas and tools with origins in topology, such as chain complexes and derived functors, to study categories of modules and other Abelian categories. Homological algebra is fairly ubiquitous, and finds applications in
module theory over rings
representations of groups and Lie algebras
sheaves on algebraic varieties
various other subjects