OpenArena is an open-source content package for Quake III Arena licensed under the GPL, effectively creating a free stand-alone game. You do not need Quake III Arena to play this game.
The Quake 3 Team Arena demo includes Single Player (with bots) and multiplayer functionality. Play the Capture the Flag, One Flag CTF, Harvester, and Overload modes with a few new skins, models, weapons, and power-ups.