Unlock Nightmare Difficulty, Special 1&2 Costume, Battle Mode, Sound Test, and Zoom Special Function for Camera by completing the game once. 

Unlock Special Costumes 3 and 4 and Fatal Difficulty by completing the game on Nightmare difficulty. 

Unlock Chapter Selection for All Difficulty Levels, Special 5 Costume, None Special Function for Camera by beating the game on Fatal difficulty. 

Unlock Sense Special Function: Complete Battle Mode. 

Unlock Chase Special Function by completing Battle Mode with an S ranking in every mission. 

Unlock Zero Special Function: Complete the Ghost List at anytime during the game then complete the game. 

Xbox Specific Ghost List:
Xbox version has 11 more ghosts than the PS2 version and listed below are where they can be found. All but the last one are hidden ghosts. The last one is available in fatal mode only and is an attacking ghost. 

Ghost - Night - Area 

Floating Twins - 1st night - Abyss
Fearful Old Man - 1st night - Burial room 
Ghost in Smoke - 1st night - Fireplace 
Smirking Man - 2nd night - Walkway (Koto) 
Woman in Mirror - 2nd night - Rope Hallway 
Wandering Child - 2nd night - Fish Tank 
Staring Ghost - 3rd night - Abyss 
Beckoning Hands - 3rd night - Demon Mouth 
Man Buried Alive - 3rd night - Forest Path 
The Sacrifice - 4th night - The Well 
Armored Warrior - 4th night - Abyss (Fatal Mode only)