In single games, tournaments, and sit'n'goes a race (race to) denotes the amount of games required to win in order to win your match. If it is a race to five, then you must win five games first to advance to win the match and advance to the next round if it is a tournament or sit'n'go. You can think of a race to two as best 2 out of 3, a race to three as a best 3 out of 5 and so on.
Games over PS$ 100 may have tables with races to 1,2,3,4, or 5 games. All games below have only the option for a race to one (winner of first game wins match).
You can check the race amount under the race column in the single games lobby, and in the tournament title or information in the tournament lobby.