In Arcade Mode, Single Race:
Completing an Area's tracks (by winning them) in Easy mode will unlock the next Area (there are six, A-F).
Completing an Area's tracks (by winning them) in Normal mode will unlock a car for you to use in Arcade Mode (two for Areas E and F).
Completing an Area's tracks (by winning them) in Hard mode will also unlock a car for you to use in Aracde Mode (again, two for Areas E and F).
Completing an Area's tracks (by winning them) in Professional mode will do sod all.
The cars you unlock are:
Area A - Mazda MX-5 89 (C-class), C-West Silvia (S-class)
Area B - Audi TT 1.8T 225 (C-class), Lancia Delta HF Integrale (R-class)
Area C - Mazda RX-8 (A-class), Gillet Vertigo (S-class)
Area D - Nissan Z Concept (A-class), Pagani Zonda Racer (S-class)
Area E - Ruf RGT (A-class), TVR Tuscan (A-class), Panoz Esperante GTR-1 (S-class), R390 GT1 LM (S-class)
Area F - Toyota GT-ONE TS020 (S-class), Mazda 787B (S-class), Viper GTS-R 02 Concept (S-class), TVR Speed 12 (S-class)
Winning a race at any track in a more difficult mode will also fill in the easier modes for that track - by racing all at Hard, all at Normal and Easy are completed for you by default.
If you wish to just unlock the tracks, race in Easy mode. The races are shorter and thus take less time.
If you complete all Arcade Time Trials at gold you are awarded a Panoz Esperante GTR-1 for use in your Sim (GT) Mode game. Or sell it - it's worth 500,000 sold.
To access Professional Mode, hold down the L1 and R1 buttons while on the mode selection (Easy, Normal, Hard) screen. "Hard" will be replaced by "Professional". In Professional mode all races are 10 laps long with tyre wear enabled. You do not get any credit for finishing Arcade Mode races in Professional mode.