The problem the Swiss system solves
A round-robin, where every player meets every other player, is only practical for small fields. Once a tournament has more than a few dozen entrants, there isn't enough time for everyone to play everyone else — the Swiss system is the standard solution, pairing players with similar scores against each other round by round instead.
How pairings are generated
After each round, players are grouped by score. Within each score group, the pairing algorithm tries to match players who haven't already faced each other, while also balancing how many games each player has had with the white and black pieces. This is why your opponents get progressively closer to your own score as the tournament goes on — by the final rounds, you're typically playing someone with the same or a very similar score.
Colour allocation and byes
FIDE's Swiss rules include strict limits on how many times in a row a player can receive the same colour, and pairing software is built to respect those limits automatically. Players who can't be paired in a round — because of an odd number of entrants at their score, for example — receive a bye, which is typically scored as a half or full point depending on the tournament's regulations.
Tie-breaks
Because Swiss pairings don't guarantee everyone with the same final score has played an identical set of opponents, tournaments use tie-break systems — such as the strength of opposition faced, or performance in games against other high scorers — to rank players who finish level. The specific tie-break order is published in each event's regulations before the first round.