It is about a square--the two-dimensional figure in textbooks--who lives in a land called Flatland, a two-dimensional land. There are descriptions on how life works in this land (gravitational pull makes rain fall from North to South; how to tell the shape of other Flatlanders, even though all you can see is a line).
That is only setting us up for the story, however. A. Square, the protagonist (the square) dreams of a place called Pointland, a concept similar to Flatland, only the citizens are only lines, exist in a one-dimensional line, and subsequently cannot travel (since it would mean going through each other). Frustrated with trying to explain the intricacies--or even the existence--of Flatland to them, A. Square gives up.
Then he is visited by a 3-dimensional figure--a sphere. I won't say any more, because that would be the whole book.
Though the story plot itself is rather short, that's not what's interesting. There are a great deal of sharp observations about the society in which Abbott lived in. For instance, in Flatland, the females are all one dimension smaller--a line, and are required by law to constantly give a warning (lest they stab to death another Flatlander), have separate exits and entrances, and are considered inferior to the males--all concepts that were prevalent in Victorian society (and may still be prevalent, but that's a different discussion). The novel also deals with class--the more sides you have, the higher up you are--thus, a circle (or a polygon with so many sides to approximate a circle) is the highest person in Flatland.
The edition I read was annotated extensively by Ian Stewart (http://books.google.ca/books?id=QkqiCVcFhZkC&pg=PA1&dq=flatland&ei=xp4YS7G3MqiOyASG9r3aCw#v=onepage&q=&f=false), and when I say annotated extensively, I mean he goes on for pages about points in the novel. There is some very fascinating math in the latter part of the book (separate from the rest of the novel written by Abbott), talking about higher mathematics (j, k, i, and n-dimensional figures).
I would recommend this book to anyone who enjoys the satire, math and can handle the prose. I tried reading this in grade six, and understood utterly nothing (otherwise I would recommend this to any child). It is a very humorous point of view, and the annotated version often goes off on a tangent, which spices up what might become a little boring. For Abbott's writing, I would say this gets a four and a half out of five.