Abbott's Flatland was a 2D world in a horizontal plane (i.e. like looking down on a map) , whereas The Planiverse is in a vertical plane, where the only available directions are left-right and up-down, with gravity applying. This means, for instance, that buildings are impossible because they would present impassible barriers; houses and other structures have to be multi-level burrows whose entrances can be jumped or bridged with a lid. Even passing other inhabitants has to be done by climbing over them, according to rules of etiquette. Some simple devices are impossible: for instance, nails are useless for fixing materials together, since in 2D a hole straight through something breaks it into two pieces. For the same reason, nothing can be porous, so there's always the danger of suffocation in any kind of enclosure. But some complex mechanisms, such as steam engines and paddled boats, and even rocket planes and a space station, can be contrived to work.
The story follows Yendred as he embarks on a religious quest, which provides a travelogue of his world that is pleasantly engrossing and full of wry humour. At one point, for instance, a female at the balloon port says to him:
You my egg to buy do want? It a beautiful blue is and very large and good to sit upon.
It's when you cross-reference this with the biology of Arde that you realise he has been propositioned by a prostitute. The Planiverse is even quite poignant in places (a couple of reviewers at the Mathematical Fiction Homepage mentioned below said it had made them cry). The Internet Archive for Math Awareness Month 2000 has a nice set of sketches from Dimensional Geographic, a project by Suttirat A Larlaub, showing Yendred's Journey through Ajem Kollosh.
If you are similarly geeky enough to seek out other fiction on mathematical themes, Alex Kasman's Mathematical Fiction Homepage is a good reference point.