I've been to the GenCon Dominion tourney for a couple of years but remember this is only what happened in the past and not how it's going to be this year.
There are 18 heats (or maybe it's 15... can't remember) on Thursday, Friday, and Saturday. Each heat consists of 18 players divided into 6 groups of 3. Each player will play 2 games facing different opponents using different kingdoms. After the 2 games they make a cut to everyone who has won twice. Since the matches are predetermined by arrival order and not swiss, it's entirely possible that the number of people have won twice would vary between 0 to 6 players. I don't think I've ever seen 0 or 6 but I remember 5 happening once. Anyway a final match is played in which the winner becomes the winner of the heat.
On Sunday all the winners of the heats gather for a more swiss style tournament. This time 5 games are played. They try to make sure you don't play an opponent or a kingdom more than once but it's going to happen. At the end of the 5 games the points (winners receive 4 for first 2 for second, in between for ties at the same place) are totaled. Usually what happens at this point is there are still 2 or 3 people who have the same point total so a last championship game is played.
The tournament is technically free, but it also shows up on the event pre-registration list. What happens then is since the tickets cost $0 people tend to hoard them and then not show up so usually if you just walk in there will be a couple of free spaces available for walk-ons. The number of free spaces actually increases as time goes on cause people will buy tickets to multiple heats, but once they've won one there's no reason to go to the others.
Oh yeah. Technically you can try the heats over again. The only restriction is that usually you can't redo the heat twice in a row cause they want to allow other people to have a chance.
Prizes vary wildly. In the earlier days it was cash. It's almost always been winner take all. More recently, if you got to the final day, you could argue them out of some dominion sets using the "Boy it sure must cost a bunch to fedex these back to Rio Grande headquarters" argument. Oh and everybody always gets the latest promo if there is a new one.