Termites generally build their nests either underground or in large timber like fallen trees.  They also build mounds, which are essentially just their normal underground nests extended above the surface.  Some mounds can reach heights of over 30 feet, but any above-ground mound is simply an extension of the subterranean tunnel system that termites thrive in.

It depends upon the species.  Some termites have a network of underground tunnels that they rely on to house their nests.  Some species create mud tubes and others live inside mounds.  Termites can also live inside the walls of a house or inside the rotten trees that they rely on for food.