There are 48 Trophies that can be collected.

BRONZE

Hypocritical
Pray 10 times

As Requested
10 tasks fulfilled

Power Drinker
Visit all 60 taverns

Treasure Hunter
Find a treasure with the help of a treasure map

Crow's Nest
Discover 10 pirate nests in a single game

Always Welcome
Attain the highest reputation in one nation

True Hero
Attain the highest popularity in a town

Peacemaker
Take over a town by peaceful methods

Conqueror
A town annexation by martial methods

Wedding Bells
Marry

Flawless Victory
Win a battle without taking any damage

Keelhaul
Rescue 100 sailors from sharks

Landlubber
Build 250 buildings

Davy Jones' Locker
1,000 vessels sunk

Hunt For Booty
Loot 25,000 barrels

Hoarding
Posess 250,000 barrels of goods in a single game

Clean Slate
Highest fame as 'Trader'

Wolf In Sheep's Clothing
Highest fame as 'Privateer'

Beelzebub
Highest fame as 'Pirate'

Arrr!
First successful boarding of a vessel

Plundering
Ambush a trader

Walk The Plank
Destroy a pirate convoy

Arm Of The Law
Destroy 10 pirate hideouts

SILVER

Treasure Chest
Save 4 million gold pieces within one game

David-Versus-Goliath
Win a battle against at least 60 guns with a pinnace

Collector's Instinct
Possess all ship types in one game

Lean Back
Patrols fought 500 battles

Lord Of War
Possess 5,000 cannons in a single game

Puppet Master
Occupy 15,000 employees in a single game

Aye Aye, Capt‘n!
Reach the highest rank

Disposable Weapons
Use up 4,000 handheld weapons in boarding combats

D-Fence
Defend a town from 5 attacks

Checkmate
Reduce a nation's number of towns down to two

Whizz-Kid
Maximize all abilities of a captain

Veni, …
Take control over 5 towns within one game

GOLD

Vidi, …
Take control over 15 towns within one game

… Vici!
Take control over 30 towns within one game

PLATINUM

Poseidon
All Trophies received

10 SECRET

BRONZE

Got Ye New Ship
Purchased first vessel

Wistfulness
First vessel sold

There, I fixed It!
Perform an emergency repair

Kudos!
Tutorial completed successfully

Welcome To The CArrrrribbean
First time at sea

Ship Ahoy!
First naval battle

Christening
Name a convoy

The Message Is Trading, Dude!
A trading route has been active for 365 days

Hey Mr. Tallyman
Turn over manually a total ammount of 100,000 barrels

SILVER

Sea Dog
Spend 50 hours in the Caribbean

http://www.ps3trophies.org/game/port-royale-3-pirates-and-merchants/trophies/