Championship mode is to Burnout 1 what the Grand Prix mode is to Burnout 3. While Race events do not amount to any progression in the game, Championships are required for progression, and they unlock new tracks and a Championship for those tracks.
Championships are a set of three Race events which carry the normal Race rules (laps, checkpoints, etc.). A Championship is usually composed of a set of three tracks that were unlocked with it. For instance, the first Championship is made up of the first three tracks available in the game, and the second Championship is made up of the three new tracks unlocked by winning the first one. However, unlike Grand Prix events, winning may not require coming in first. Instead of earning points for placings, the player has a minimum placing they have to come in at the end of each race in order to continue to the next race. Through the course of the game the minimum placings become higher and higher until all the races require placing in first place to continue to the next race.
If the player does not come in the minimum placing (or higher), they can use up to three tokens to try the race again. Tokens are carried over into new races in the Championship, however the player cannot earn more, so they have three tokens for the entire Championship. This, coupled with the unforgiving nature of Burnout 1, means that Championships can be quite tough to complete.