Backgammon tournaments are great fun, and they are the best method of sharpening your game. If you have never played in one, you should do so as soon as possible. Find out if anyone is running Backgammon tournaments online (look at the left side). I think you will find that from then on you'll play in them whenever you can.
Here is how a Backgammon tournament is usually run. It is a knock-out event, and the draw is made in the same way as for any other such event, e.g. a tennis tournament. Each round is played up to a certain number of points. Normally the number of points you need to win increases in the later rounds. So in a Backgammon tournament of 64 people the early rounds might be up to 5 points, the Semifinals up to 7 points and the finals up to 9 points. In a major Backgammon tournament lasting more than one day, and probably four days, the number of points needed to win in each round is much higher.
Doubling is permitted. Gammons and backgammons count double and triple as usual. But one change from the normal rule about doubling is now commonly played. This is the Crawford Rule (invented by the late, great American bridge and Backgammon player, John Crawford), which stipulates that when a player has reached one point short of the number needed to win his match, his Opponent may not double in the next game. So, if you were playing a match up to 7 points and your Opponent reached 6 points, you could not offer him a double in the next game. After that game the double is free again.