Much like a Round Robin tournament, players do not get eliminated in the middle of a Swiss tournament. Unlike a Round Robin tournament, not every player will play every other player in the competition.
In a Swiss tournament, players will be randomly paired with another player with the same record as them in every round. This means that in Round 1, you could be paired up with anyone else in the tournament. However in Round 2, you'll be paired up against someone else with the same record as you (0-1 or 1-0). This will continue on until all rounds of the tournament have been played.
Often, a Swiss bracket will be followed by an elimination bracket with the top placing players from the Swiss bracket being seeded. Swiss tournaments are very popular in card games and board games. If you're running a tournament for Hearthstone, Magic Online, Pokémon or another game with more variance in results, a Swiss tournament is something your players are likely interested in! Swiss is also useful in casual tournaments for giving all players many games as nobody gets eliminated -- especially as players will be playing those of similar skill to them as the tournament goes on.
Watch this tutorial we've made on Swiss Brackets: