A while ago, Id Software released the source code for the Quake 3 engine. They did not, however, release all the maps and models under a free license, so at the time, you still had to buy the game to play it. No more! Thanks to the OpenArena project, there's a cross-platform Quake client with free models and maps.
Download OpenArena here, extract all the files, and you're set. Sort of. The OpenArena website only hosts version 0.7.0, a buggier version. A patch for version 0.7.1 is available here, which fixes the constant crashing on Fan- a real nuisance.
OpenArena is like any other FPS. You run around shooting bots and collecting ammo. The scroll wheel cycles through weapons, and holding down the middle mouse button gives you an optical zoom on any weapon. Simple enough.