WELCOME TO CFC!
Ahh, I got to be the first to say it. Anyway, there are several different ways to play multiplayer.
1. Hotseat- this is where you and a friend or friends play the game on one computer. You take a turn, your friend sits down for his turn, and you keep rotation.
2. PBEM- also known as Play By E-mail. This is the forum for PBEM so that's what you will find here. The way this works is you take yoru turn, end your turn and save the game. Then you send that save file to your friend, he loads it, plays his turn, ends his turn and saves it, then e-mails it back to you. As you can probably tell, this makes the games last a long time.
3. Direct connection- this is where you play with people directly over the net in real time. The games are usually laggy and slow going, but you can usually play one in about 2-4 hours. Some are shorter, some are longer.
All these games are set up from the main screen when you start Civ3. Click on the tab called multiplayer. There you will make a choice on how to play, either over the net, hotseat, PBEM. Pick which one you want. Then set the game up however you want, hit the launch button, and play. It's really pretty easy. If any of these instructions are unclear, let me know and I can try to talk you through it better.