The original Burnout naturally contains the core events for all Burnout games, however it does contain it's own share of unique events. Here is a full list of them.
The Race event is a competition between you and three racers to go through Checkpoints to keep the timer going. Most tracks are three lap loops, however some are routes made up of portions from multiple tracks.
Championships are the main form of progression in Burnout 1. They are a compilation of three Race events, usually on a set of three tracks that have been unlocked. Instead of points for placings like in every other Burnout game, the player has to finish at a specific placing for each race to win.
Face-Off events are a special race event between you and an opponent in a special car. If the player wins a Face-Off, they win the car the opponent was racing in for use in other events (even the very same Face-Off).
The unlockable Survival mode is a solo event where one crash ends the event, no mean feat. There is no reward for completing all tracks in Survival Mode
Free Run Mode is the opposite of Survival Mode, where the lone player can play through the tracks with absolutely no traffic. Free Run Twin is a similar mode, but it's played with a second player.