There are 12 Trophies that can be collected.

Bronze

Arcade C-license
Get C class license in Arcade mode.

Arcade B-license
Get B class license in Arcade mode.

Arcade A-license
Get A class license in Arcade mode.

Arcade S-license
Get S class license in Arcade mode.

Original C-license
Get C class License in Original mode.

Original B-license
Get B class License in Original mode.

Original A-license
Get A clss License in Original mode.

Original S-license
Get S class License in Original mode.

Silver

Skillful driver
Clear 1-1 through 1-S of Crazy Box.

Superb Driver
Clear 2-1 through 2-S of Crazy Box.

Amazing Driver
Clear 3-1 through 3-S of Crazy Box.

Gold

Crazy Driver
Clear Special 1 through Special Stage of Crazy Box.

http://www.ps3trophies.org/game/crazy-taxi/trophies/