There are 13 Trophies that can be collected.

Bronze

Check    
Put an opponent in check in an exhibition game.    

Checkmate    
Win an exhibition game against the computer.    

Amateur    
Complete all the tutorials.    

Pure Correspondence    
Win a game of Play By Mail.    

Piece Collector    
Capture 100 pieces.

Silver

Tournament King    
Complete Tournament mode on all three difficulties.    

Problem Wizard    
Solve 100 mate problems.

Gold

Grandmaster    
Achieve an ELO of 2500 or a Grandmaster Pure Chess rating.

5 Secret

Bronze

Cross-check    
Play a check in reply to a check.    

Promotion    
Advance a pawn to the eighth rank in an offline game.    

O-O-O    
Perform a castling move on the queenside during any full game.    

En Passant    
Perform an En Passant during any full game.

Silver

Pion Coiffé    
Deliver checkmate with a pawn.

http://www.ps3trophies.org/game/pure-chess-vita/trophies/