Ultimate Frisbee, explained as simply as possible, is a game played between two teams of seven players on a large rectangular pitch. A line drawn across the pitch at either end creates two “endzones” (like in American Football, or Rugby). These are the goal-scoring areas. A goal is scored when a team completes a pass to a player standing (or more likely running) in the endzone they are attacking.
Players cannot run with the disc. When you get the disc you must come to a stop and try to throw it to another player (a bit like netball). By passing from player to player, the offence attempts to work the disc up the pitch towards the endzone they are attacking. If the disc hits the ground or is intercepted or knocked down by the other team, then the opposition takes possession (it's called a "turnover"). Possession also changes if a receiver is outside the playing area when he or she catches it.
The defending team attempts to stop the team with the disc from making progress upfield by marking them (as in soccer or basketball). The theory is that the offence won’t want to pass to a player who is being marked closely, as it’s likely to result in an interception. So it boils down to the offence players trying to get free of their markers to receive a pass, while the defence makes every effort to stay with them in the hope of forcing a turnover.