Attempt this early on in the game to make it easier and restart the race if you mess up. If you have already progressed through the game some, you may want to visit the quick play section from the main menu for an easy track.
You will have to get to the finish line before the cops fill up the red bar on your HUD. You start out being chased by only one cop, but gradually add more up to a total of three throughout the race. Try to make good turns and to avoid getting boxed in at all costs, as it will quickly lead to a game over.