Visual group theory
Nathan Carter's book Visual Group Theory tries to help you understand group theory through a set of pictures. Some of these made sense to me while others didn't. Here's the illustration of the structure of dihedral groups from a presentation that Carter gave before the book was available. That seems to show their structure in a very useful way.
Carter's book runs 297 pages, many of which are illustrations like this one. It's a book that I would have found very useful back when I was taking classes in group theory. It would have definitely helped to make some concepts clearer. If you're interested in group theory, this is probably a book that you'll want to have a copy of.
Carter has also developed software that lets you create visualizations of groups. This is Group Explorer, and it's something that I wish I had time to actually try. Maybe I'll do that on Voltage's next Jack Bauer Day.