Good book on homology

There is an algebraic topology book that specializes particularly in homology theory-namely, James Vick's Homology Theory:An Introduction To Algebraic Topology. It does a pretty good job of presenting singular homology theory from an abstract,modern point of view, but with plenty of pictures. It's an underrated book for this purpose that I wish would get more attention-you may find it very useful.

Another book I think you'll find very useful is the aforementioned Joseph Rotman's An Introduction To Homological Algebra. Not only is it the most accessible and clearly written book on the subject, it emphasizes the topological origin of many of the central concepts.

I think both of those,in addition to the other good suggestions here,will help you out a lot.

As mentioned in the comments Peter May's Concise Guide is an excellent book.

Some others to consider:

• Algebraic Topology - Tammo Tom Dieck. Apart from being an excellent text book (see my answer here for example) this has a heavy use of categorical language (already on page 20 he is talking about functors and left adjoints) so if that is your thing, this is the book for you!.
• An Introduction to Algebraic Topology - Rotman. This has a gentle use of categories and functors.

If you're after something purely homological in nature (i.e. without reference to the applications of (co)homology to topology), then the standard reference would be Weibel's book An Introduction to Homological Algebra.. This is very categorical, but it isn't specifically about homology and cohomology in topology.

If you're looking for something more directly related to (co)homology of spaces, then I'd like to recommend Switzer's book Algebraic Topology - Homology and Homotopy. It has a nice treatment of homology and cohomology from the categorical perspective. He takes the approach of first discussing general (co)homology theories on the category of spaces, and then even goes through Brown representability before turning to singular (co)homology. In fact, I'd recommend this book as a wonderful alternative to Hatcher if you find his geometric arguments and lack of category theory unsatisfying. Switzer seems to me to be more rigorous (less pretty pictures), but he uses more category theory and I prefer the rigour.