While I don't know of any PbP Eclipse Phase games off the top of my head, I know RPG.net has a section on PbP games -- you can read the recruitment thread, the IC thread, and the OOC thread for any number of different PbP games there. Essentially, though, it's just like any other text-based form of gaming, except much slower. The GM sets up a scene, the players take turns reacting, and the GM responds accordingly. What might take a single night of gaming at the table can easily take a month or more, depending on how often people are posting. I've found they typically have a higher attrition rate than tabletop or chat-based games, although that's going to vary based on the group.