Tournaments on Ultimate Golf come in several varieties, but there typically only one type of tournament going on at any given time.
The current schedule (which is subject to change) typically consists of:
Mon, Wed, Fri: Shootout Tournament - Three hole tournament with inexpensive coin entry fee, unlimited premium currency replays and huge prizes.
Tue, Thu: Nine-hole Tournaments
Sat, Sun: Eighteen-hole tournaments
Each tournament has multiple levels - currently Amateur, Rookie and Veteran, but with Pro and Tour Pro levels in the works sometime in the future.
The level of the tournament impacts the entry fee, the prizes awarded, the size of the group that you'll compete against and factors such as wind speed and hole and tee position.
Tournaments can be played at your own pace. If you end your session after completing a hole, you can pick up the tournament round where you left off...no need to play the entire thing in one sitting.
Also, you can purchase replays, which are additional entries in the same tournament.