How it works
Teams are paired, the winners advance, and the losers go home. Repeat until one team is left. Each round halves the field, so 16 teams play 4 rounds: round of 16, quarterfinals, semifinals, final.
Why n − 1 games
Every game eliminates exactly one team, and all but one team must be eliminated. So there are n − 1 games for any n: 7 for 8 teams, 31 for 32, 5 for 6.
The third-place game
The two semifinal losers can play once more. It adds a game, so 16 teams play 16 games instead of 15. It is optional and most useful when there is a prize or medal.
The weakness
One bad game ends a team. That is fine for short events and large fields. If you want teams to play at least twice, use double elimination or pool play.
Questions
What if I have 6 teams?
The bracket has 8 slots, so 2 teams get byes. There are 5 games in 3 rounds.
Is a consolation bracket worth it?
It gives first-round losers another game. It does not decide the title, so run it only if you have courts and time free.