KO system

This is the direct elimination system. A player who loses a game leaves the tournament. The tournament continues until one single player stay and that player is the winner. For a tournament with 2ⁿ players, n rounds will be necessary.

RoundRobin system

Each player meets every other one.

Swiss system

Comes from the KO system. But there is no elimination. Pairing is made between players having the same number of wins.
To work correctly, a minimum of n rounds is required for 2ⁿ players. This is sufficient to clearly determine a winner. However, in order to be able to set a satisfying placement for all players, n+1 or n+2 rounds are recommended.
For final placement a tiebreak method is usually implemented.
See also the rules for inner-group pairing.
If there are more players in a tournament than the number of rounds can handle (to determine a clear winner), then separating players in categories can be considered.

Accelerated pairing

If there are more players in a category than the number of rounds can handle, then "accelerated" pairings can be considered. Accelerated pairing is popular in chess tournaments. With such a system, it is possible to handle a 2ⁿ⁺¹ players tournament with only n rounds. If the initial seeding is reliable, only one player will achieve the tournament with n wins and be therefore a clear winner of the tournament.

Mac-Mahon system

It is the most popular system in European Go tournaments
See a description of Mac-Mahon System