This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question would be the name of such a text. (At this point, however, I would strongly prefer a good intuitive explanation to a rigorous description of the modern theory. It would also be nice to get some picture of the historical development of the subject.)
Some sub-questions: what does the condition that d^2 = 0 means on an intuitive level? What's the intuition behind the definition of the boundary operator in simplicial homology? In what sense does homology count holes? What does this geometric picture have to do with group extensions? More generally, how does one recognize when homological ideas would be a useful way to attack a problem or further elucidate an area?