I have a fair understanding of homotopy: it's the study of homotopy groups, which are groups formed by different ways (up to smooth deformation) to map a sphere to a space. More abstractly, it's about morphisms from the pointed sphere to a pointed topological space.
Homology from what I gather is what you get from turning a homotopy group to an abelian group. Since a homotopy group is not abelian, this cannot happen painlessly, and so homology is quite different from homotopy.
Homology groups are abelian, and much more amenable to symbolic manipulation, allowing people to study topology without having to draw pictures. This gives rise to homological algebra, the primary tool of algebraic topology.
Homological algebra from what I gather has great use outside algebraic topology, although I have no idea why.
Cohomology algebra is already way out of my understanding. It is obtained by reversing all the arrows in homological algebra diagrams, and then interpreting them in some way that makes them very valuable. The only cohomology I know is de Rham cohomology, and that basically describes how a space can have locally exact differentials that are not globally exact. The most basic example is the plane with origin missing, then the 1-covector field $d\theta$ is locally exact, but not globally exact.
So that leaves cohomotopy. What is it good for? I can't find any introductory material or overview on its applications.