With the current version of Civ III you cannot play multiplayer games. The closest things to multiplayer that you can do are "succession" and "democracy" games.
A succession game is where you play several turns of a game, then email the file to someone who plays another few turns, etc... Usually this is done in groups of 4 or 5.
The democracy game is only currently running for Civ II, but basically a whole bunch of people vote on what should happen. The majority rules, and their decisions are implemented by the president.
Of course, you're not playing against other people, so it's not really multiplayer -> more of a co-operative effort.