A lot of the most exciting developments in contemporary physics involves discovering the structure of space-time. The introduction of String Theory, and more recently, M-Theory into the public consciousness has raised the issue of whether or not higher dimensions exist. The question for physicists though is not if, but how many. Although it may be fashionable to drop words like "p-brane", and "graviton" at a party (if not fashionable, at least it's geeky), few people even want to think about how higher dimensions might work. We just don't have an innate sense of anything beyond our three dimensions of space and one of time with which we are familiar.
Fortunately, we can use our imaginations, and a fantastic book written way back in 1884. Flatland: A Romance of Many Dimensions, was written by Edwin A. Abbott an English clergyman and teacher. This classic of science fiction is a thin little book but its an easy and quick read. It is also a cheap read. The edition published by Dover is a mere $1 USD.
The story is a social reflection of the time it was written, but it provides a wonderful setting to explain what a two dimensional world (the Flatland of the title) is like for the main character, A. Square. Later, A. Square is taken up from his plane existence, and shown other worlds by an inhabitant of Spaceland. He is shown Pointland and Lineland, as well as Spaceland. So blown away by a revelation about the existence of other dimensions is he that he begins to wonder about even more dimensions. The explanations of the different "lands" are clear and full of useful examples that help the reader understand the expansion of dimensions.
Several sequels have been written by other authors, but the original remains the best.