There are 40 Trophies that can be collected. 

Bronze

The Curse of the Black Pearl
Complete the Film 1 story.

Dead Man's Chest
Complete the Film 2 story.

At World's End
Complete the Film 3 story.

On Stranger Tides
Complete the Film 4 story.

Here there be monsters
Get eaten by a creature in deadly water.

A weather eye on the horizon
Use a spyglass.

The Green Flash
Watch a sunset.

Welcome to the Caribbean!
Complete Port Royal.

Hello, poppet!
Unlock all Elizabeth characters (Single Player Only).

The best pirate I've ever seen
Complete Port Royal in Story without dying.

The worst pirate I've ever seen
Complete Port Royal in Story with zero studs.

More what you'd call guidelines
Complete the Brethren Court.

A pirate's life for me
Test any custom character.

You filthy, slimy, mangy cur!
Complete all the Guard Dog levels.

Try wearing a corset
Do 5 lady backflips in a row.

Savvy?
Unlock all the Jack Sparrow characters (Single Player Only).

Wind in your sails!
Hit a flying parrot on Smuggler's Den.

Did everybody see that?
High dive into the Maelstrom.

Aye-aye, captain!
Play a level in co-op.

Gents, take a walk
Walk on the sea bed with all possible characters.

I am a bad man
Play a level with all Extras turned on (Single Player Only).

Fight to the bitter end!
Defeat 100 enemies.

Five lashes be owed
As Jimmy Legs, whip Will Turner 5 times.

Fire!
Fire 100 cannonballs.

You're off the edge of the map
Highlight the secret 6th point on all 4 level select maps (Single Player Only).

Silver

Believing in ghost stories
Unlock all the cursed Black Pearl crew characters (Single Player Only).

Do you fear death?
Unlock all the Flying Dutchman crew characters (Single Player Only).

The pirate all pirates fear
Unlock all the Queen Ann's Revenge crew characters (Single Player Only).

You may throw my hat
Collect all the red hats (Single Player Only).

The Brethren Court
Unlock all the Pirate Lord characters (Single Player Only).

What do you want most?
In any level use only the compass to find all it's secrets in one go, alone.

Parley!
Unlock all characters (Single Player Only).

And really bad eggs
Play as all the Extra Toggle characters.

Pieces of Eight
Reach 888,888,888 studs.

There's the Jack I know
Get True Pirate in all levels (Single Player Only).

Sea turtles, mate
Ride on all types of animal in the game.

Hello, beastie
Get eaten by the Kraken 10 times.

Gold

Take what you can
Collect all Gold bricks (Single Player Only).

Now bring me that horizon
Complete the game to 100% (Single Player Only).

Hoist the colours!
Sail all the minikits in the hub.

Platinum

All hands on deck!
Collect all Trophies.

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