There are 16 Trophies that can be collected.

Bronze

Rescue    
Rescue Yorda    

Failure    
Finish watching the demo scene confronting the Queen at the front gate stage

Silver

Armed and Ready    
Acquire Sword    

East Gate    
Open East Gate    

West Gate    
Open West Gate    

Farewell    
Finish watching farewell demo with Yorda at the front gate stage

Gold

Royal Arms    
Acquire Queen's Sword    

Emancipation    
Clear the game    

Split the Watermelon    
Bring the watermelon to Yorda upon completing 2nd time through    

Spiked Club    
Acquire spiked club    

Shining Sword    
Acquire shining sword    

Bench Warmer    
Save at all save points    

Express Journey    
Beat the game within 4 hours    

Castle Guide    
Beat the game within 2 hours    

Unscathed Escape    
Clear the game without viewing a game over screen

Platinum

Enlightenment    
Acquire all trophies

http://www.ps3trophies.org/game/team-ico-collection-ico/trophies/