A somewhat bizarre entry, but wanted to include the nlab discussion: http://nlab.mathforge.org/nlab/show/cohomology
Could try to answer some of the following questions one day:
http://mathoverflow.net/questions/6125/what-is-a-cohomology-theory-seriously
http://mathoverflow.net/questions/27121/k-weil-cohomology-theories
http://ncatlab.org/nlab/show/cohomology+group
nLab page on Cohomology