Most of us here play Play By Forum (PBF). Basically we make a move and post a detailed description of the move in a forum. Then our opponent reads our move description and makes their own moves and posts them. And thus the cycle continues. This often involves attaching game files such as current board maps etc.
Others play Play By Email (PBEM). This involves emailing back and forth instead of posting on a forum. This often involves emailing game files as well.
Some people play live online (networked gaming). There are a few different games out there that allow this, but one free one can be found at triplea.sourceforge.net.
Take a look in the Games In Progress forum to see how we play here.