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The principle of a Swiss tournament is that each player will be pitted against another player who has done as well (or as poorly) as him or herself.
The first round is either drawn at random or seeded according to rating. Players who win receive a point, those who draw receive half a point and losers receive no points. Win, lose or draw, all players proceed to the next round where winners are pitted against winners, losers are pitted against losers and so on. In subsequent rounds, players face opponents with the same (or almost the same) score. No player is paired up against the same opponent twice however. In chess it is also attempted to ensure each player an equal number of games with white and black, alternate colors in each round being the most preferable, and definately not the same color three times in a row.
The detailed rules of how to do the pairing are usually quite complicated and often the tournament organizer has access to a computer to do the pairing for him. If the rules are strictly adhered to, the organizer has no discretion in pairing the round. See the link below for detailed pairing rules from FIDE.
The tournament lasts for a number of rounds announced before the tournament (usually between 3 and 9 rounds). After the last round players are ranked by their score, if this is tied a tie break score (such as the sum of all their opponents' scores) can be used.
Determining a clear winner (and incidentally, a clear loser) usually requires the same number of rounds as a knockout tournament, that is the 2-logarithm of the number of players rounded up. Therefore 3 rounds can handle 8 players, 4 rounds can handle 16 players and so on, however it is not uncommon to have more players than this, and as a result it can happen that two or more players finish the tournament with a perfect score, having won all their games but never faced each other.
Compared to a knockout tournament the Swiss system has the inherent advantage of not eliminating anyone. That means that a player can enter such a tournament knowing that he will not be knocked out in the first round and not get to play any more. The worst that can happen in this respect is being the player left over when there is an odd number of players. The player left over receives a bye, meaning he does not play that particular round but receives a full point as if he had won a game. He is reintroduced in the next round and will not receive another bye. A Swiss system tournament does not always end with the exciting climax of the knockout's final however. Sometimes a player may have picked up such a great lead that by the last round he is assured of winning the tournament even if he loses the last game.
Compared with a round robin tournament, a Swiss can handle many players without requiring an impractical number of rounds. However the final rankings of a Swiss tournament are usually more random. Even though the correct player usually wins, and the correct player usually winds up in last place, the players in between are only sorted roughly.
A variant known as the McMahon system tournament is the established way in which European go tournaments are run. This differs mainly in that players start at different levels; so the Swiss system is the special case where all players start at the same level. It is named for Lee McMahon (1931-1989) of Bell Labs.