In double elimination tournaments, teams are not eliminated from the tournament until their second loss. The way this is achieved is with a losers bracket.
Everyone starts out on the top bracket, or winners bracket. After the first round, when half the teams lose, they drop down to the loser bracket. A loss in the losers bracket eliminates you from the tournament. In the next round, half of the remaining teams in the winners bracket lose, and they go on to play the winners of the first round of the losers bracket. At the end of the tournament, you have two teams left: one is the winner of the winners bracket, who has never lost, and one is the winner of the losers bracket, who has one loss. They play each other; if the winners bracket team wins, the tournament is over, but if the losers bracket wins, they play again, as now each team has one loss.
Just as in a single elimination tournament, if you don't start out with a number of teams that is a power of 2 (2, 4, 8, 16, 32, 64, etc.), then some teams will need to play an extra game, and others will get a bye in the first round.
Let's look at the 9 team double elimination bracket that you posted. Since 9 is not a power of 2, we'll need an introductory round where only 2 teams play, and the other 7 teams get a bye. The 8 and 9 seeds play each other in game 1; the winner goes on to play the number 1 seed in game 5, and the loser becomes team "L1" and plays the loser of game 2 (either the 2 or 7 seeds).
You can see what happens for each game on the bracket; the winner of the game goes on, and the loser of the game drops down to the loser bracket in the "L" spot for whichever game number they lost on.
In the end, the winner of game 13 and the winner of game 15 (which is a losers bracket game) play each other in game 16. If the losers bracket team wins, the two teams play again in a 17th game.
Compared to a tournament with 8 teams, where every team is on equal footing when the tournament starts, in a 9 team tournament, Seeds 8 and 9 are at a disadvantage, because they have to play an extra game at the beginning, where every other team gets a bye. Seeds 2 and 7 are also at a disadvantage if they lose their first game, because they will have an extra game to play in the losers bracket.