A Swiss tournament is similar to a Round-Robin tournament in that no players are eliminated. Every player will play every round, and the player with the highest number of points at the end of the tournament is the winner.
How does a Swiss tournament work?
In the first round only, high rated players are intentionally paired with low rated players. After the first round, pairings are based on performance: those who won are matched with other winners, losers with losers. There may be some exceptions to avoid color repetition.
In a Swiss tournament you don't necessarily play every other player. If there are a lot of players in the tournament, there may not be enough rounds for everyone to play everyone.
Swiss Scoring System
Loss: +0 points
Draw: +0.5 point
Win: +1 point
Possibility of a 1 point "bye" if there are an odd number of players, normally given to the player lowest in standing. Read more about byes here.
You can learn more about Swiss tournaments by visiting the Wikipedia page.