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Successfully complete the indicated task to unlock the corresponding car:

Assassin Compact: Get 15 Takedowns.
Compact Collector: Win the Gold medal at the Silver Lake Eliminator (type varies in each game).
Compact Cop: Win the bronze medal or higher at the Silver Lake or Airport T1 and T2 Pursuit.
Dominator Compact: Get 10,000 Burnout Points.
Gangster Legend: Win the gold medal at the Palma Bay Legend Face-Off.
Legend Compact: Win the gold medal at the Interstate Loop Legend Face-Off.
Tuned Compact: Win five gold medals.

Successfully complete the indicated task to unlock the corresponding car:

Assassin Coupe: Get 60 Takedowns.

It may seem odd but in Pursuit mode if you get in front of the car your chasing try crashing, if you manage to direct your car into the front of your rival it will often result in a takedown even the cars health bar is full.

To unlock everything, type in "BURNOUT" as your profile name. You will have all cars and such.

Password - Effect

BURNOUT - Everything Unlocked

Unlock all Legend Face-Off cars to unlock Mirror mode in Single Event mode.