I’m not highly familiar with Swiss tournaments but I understand that in such tournaments everyone plays everyone else one at a time.
There are basically three types of tournament.
- Knockout (& double elimination)
- Round-robin (& double robin)
- Swiss (& McMahon)
A knockout tournament is the classic pyramid shape, where each time you win a game you move into the bracket of people who won in the previous round. This is practical for when there are a middling number of people, let’s say 8 – 64. A double elimination knockout uses a second pyramid as well, designed to give everyone who loses before the final a second chance via another route.
A round-robin is what you’ve described, where every competitor plays against every other competitor. This is practical for a small amount of people, generally less than ten. A double round-robin is the same thing, but you all play each other twice – this is used so that every encounter happens with both colour combinations.
A Swiss system is what you use when you have quite a lot of people, and you don’t have time for many rounds. There are two types of Swiss system tournament: the basic Swiss and the McMahon. In a basic Swiss, every player starts with a score of zero and is paired randomly. As the tournament goes on, players are matched against opponents with the same score. This system is used by professional tournaments like the Chess Olympiad. Your basic amateur Go tournament, though, uses the McMahon system. In a McMahon tournament, players begin with a score based on their rank – this prevents having a few mismatched rounds at the beginning.
PS. “the bar” in a Swiss system tournament is the lowest rank at which, if you finish with the highest score, you’re counted as having won (because in the majority of tournaments there isn’t enough time for the stronger players to build up a natural lead.) In my experience in the UK, the bar is usually set at 2k – 1d.