In short, The Game divides a class into teams who compete against one another for points. The team with the most points at the end of a Unit wins and gets to stay together while the other teams are disbanded to various degrees depending on how well they did. New groups are then formed. The goal is to try to have your team win more Units than any other team.
The is a LOT to The Game. I've written four pages and haven't even come to the various adversities that the teams can face (or inflict upon one another). Still, I should get it written down at some point, and this current writing bug that I seem to have is the perfect excuse to do so.