There are 50 Trophies that can be collected.

Bronze

MAIN OPS All Rank A    
Cleared all MAIN OPS at Rank A or higher.

EXTRA OPS All Cleared    
Cleared all EXTRA OPS missions.    

EXTRA OPS All Rank A    
Cleared all EXTRA OPS missions at Rank A or higher.

Fulton Recovery Veteran    
Reached 300 Fulton recoveries.    

Hold-Up Artist     
Reached 100 hold-ups.     

Lethal Enforcer    
Reached 100 kills.    

Stun Specialist    
Reached at least 100 stun attacks or sleep-inducing attacks.    

Costume Collector    
Obtained every costume.

Hideo Kojima    
Recruited Hideo Kojima.    

Mech Catcher    
Captured a mech.    

Paz's Thoughts    
Read all of Paz's diary.    

Huey's Thoughts    
Read Huey's love letter.    

Strangelove's Thoughts    
Read all of Strangelove's memoirs.    

Monster Hunter    
Cleared all hunting quests.    

CQC Six Shot    
Performed CQC six times in a row.    

Leap of Faith    
Performed a Leap of Faith.    

I feel… safe    
Wearing a cardboard box gives you that feeling of inner peace.  

Indirect Attack    
Attacked enemy soldiers by blowing up an oil drum or truck (single player only).    

OUTER OPS Sortie    
Carried out a mission in OUTER OPS.

Trader    
Carried out a trade.    

Delivery    
Sent an item using Delivery.    

Paz Mania    
Obtained Rank S on a date with Paz.    

Kaz Mania    
Obtained Rank S on a date with Kaz.    

That looks tasty!    
Nicely cooked a ration.    

VERSUS OPS All Rules    
Played using all of the rules in VERSUS OPS.    

Divine Wind    
Collected a soldier using a divine wind.

Silver

MAIN OPS All Rank S    
Cleared all MAIN OPS at Rank S.

EXTRA OPS All Rank S    
Cleared all EXTRA OPS missions at Rank S.

Special Staff Recruiter    
Assembled all special staff members.

Elite Forces    
Cleared 'Capture the Enemy Base 06' in OUTER OPS using only soldiers.

Gold

Big Boss    
Cleared all MAIN OPS and EXTRA OPS missions at Rank S.

Platinum

Vic Boss    
Obtained every trophy.

18 Secret

http://www.ps3trophies.org/game/metal-gear-solid-peace-walker-hd/trophies/