There are 4 boss cars to unlock in the game:

Tank: 1969 Dodge Charger
Triple Dub: Subaru Impreza WRX STi
DJ Wes: Chevrolet Corvette Z06
GT King: Nissan Skyline R34 GTR

These cars are unlocked by beating them in Boss Races and are placed in a special location in your Pocket Garage.

The boss car the game chooses to race against you depends on the level of the race and the car you are using. You need to defete the same boss car in multiple different Boss Races in order to unlock that car. You can choose from any of the Boss Races in Circuit Mode and Street Cross and Drag from Quick Play.