Ninety-six players, five rounds, one hall, and by Sunday evening somebody will be on four and a half points. Nobody gets sent home on Saturday morning. That is what a Swiss buys you, and it is why almost every weekend open you will ever enter is one. The system is named after Switzerland, where it was first used at a tournament in Zurich in 1895.
Round one: the top half plays the bottom half
The arbiter sorts the entry list by rating and cuts it down the middle. With ninety-six players, number 1 gets number 49, number 2 gets number 50, and so on to 48 against 96. So the 2100 draws the 1550 and the 1900 draws the 1400. Round one is meant to be a formality for the favourites, and mostly it is.
After that you play your own score
Round two splits the field into score groups: the players on 1, the players on a half, the players on 0. Inside each group the same top-half against bottom-half cut happens again. Win and you go up a table, lose and you go down, and after three rounds you are playing people who have had the same kind of day as you.
Two constraints bind the pairing. You never play the same opponent twice, and colours are kept as even as possible, so you should not get three whites in a row and over five rounds you expect three of one colour and two of the other. No arbiter does this by hand any more. They run the FIDE (Dutch) system in pairing software, and the rules behind it fill section C.04 of the FIDE Handbook.
If an odd number of players turn up, somebody is left over and gets a bye instead of a game.
Why it is not a round robin and not a knockout
A round robin is everyone against everyone: fourteen players, thirteen rounds, which is what the Tata Steel Masters at Wijk aan Zee does every January. It is the fairest format there is and it does not scale. Ninety-six players would need ninety-five rounds.
A knockout is the other extreme. Lose your mini-match and go home; the FIDE World Cup starts with more than two hundred players and halves the field every round. Quick, dramatic, and one bad morning ends your fortnight.
The Swiss sits between them. Everybody plays every round, and the pairings do the sorting that a knockout does with the exit door.
Tiebreaks, which is where the prize money goes
Three players finish on four and a half out of five and there is one first prize. Points cannot separate them, so the entry conditions name a tiebreak, decided before the event and not after. The usual first one is Buchholz: add up the final scores of every opponent you faced. Beat five players who all did well and your Buchholz is high; the same score against a soft draw scores lower. Then come Sonneborn-Berger, the direct result if the tied players met each other, and sometimes the number of wins.
I read that line on the entry form before I enter anything, along with the number of rounds and the time control. And then enter one. Nine rated games against strangers will teach you more about your own weaknesses than a month of online blitz, which is the argument I make at more length in the piece on breaking a rating plateau.