Real quick history: "Personal Computers" are computers that were personal because they did their own processing so a person did need to be hooked up to some mainframe to use a computer. Pretty much when we use a computer these days we use a PC. But back in the day we would sit at a "dumb terminal" which was simply a keyboard and a monitor connected to the mainframe computer. Thus, all the word processing or spreadsheet work we were doing at the dumb terminal was actually being processed at the mainframe.
Thin clients are more similar to dumb terminals than to PCs. In a thin client network you have one server often hooked up through a fast connection to a network switch, and many thin clients (i.e. not "thick clients") hooked up to that switch through fast, though often not as fast, connections to the network switch.
The Ubuntu server, for example 8.0.4, can act like a regular server on a network with thick clients (i.e. PCs). However, it also has the capability to be used as a thin client server. Ubuntu 8.0.4 uses LTSP (Linux Terminal Server Project) 5.0. Here are directions to set up the HardyClassroomServer.