Quake 3 Arena is what is called an 'Arena' or 'Tournament' style First Person shooter. Quake 3 is basically a fast paced arcade style internet death match game. Capture the flag is also available, and there are many unofficial mods created by the public available for download.
Quake 3 is more than just first person shooters stripped down to their bare bones. The engine is legendary, and is still featured in games today such as Call of Duty; the graphics still look decent in the year 2003. The gameplay is what really shines however.
The beauty of Quake 3 is the balance, something found in a select few games. Most of the balance is in the weapons; how even the machine gun (pea shooter) you start with can be very effective at killing. There is a super weapon, the BFG, but it is often disabled on servers. Even if it is not, the BFG is not featured on all maps.
Part of the attraction of Quake 3 is the challenge. Today the net is filled with pro Quake 3 players that have been playing since the game was released; however, if that intimidates you too much, you're welcome to play against the bots. The game scales well with different skill levels, and there is no such thing as perfection with the game's loose physics engine. Trick jumping and other physics 'exploits' are one of the main attractions to experts of the game.
The game combines killer technology (for several years ago), with careful gameplay balance to make a transcendent experience.