Single elimination
Players are paired in a bracket. Lose once and you are out; the last player standing wins. Fast, and the number of rounds grows only with the log of the player count.
Tournaments
Round robin
Every player meets every other player once. Points decide the ranking. Takes longer, but every client is tested against the whole field.
Live tournaments
Single elimination Tic-tac-toe
Tic-tac-toe · 8 players · semi-final
Round robin Othello
Othello · 5 players · match 6 of 10