Here is the full list of all 51 Port Royale 4 (EU) trophies - there are 38 bronze, 11 silver, 1 gold and 1 platinum.
Raid a trader
Buy a letter of marque
Complete the Spain campaign
Build a cathedral
Find your first treasure
Accumulate 1,000,000 gold
Sell 1,000,000 barrels in one game
Host a large feast in the church
Sink a pirate convoy
Manage 3 towns in one game
Collect all captain's tactics in one game
A trade route has been active for 365 days
Sell a vessel for the first time
Construct 50 town buildings in one game
Produce 50,000 barrels of goods in one game
Complete your first town's task
Name a convoy
Hire a captain
Unlock your first concession
Buy a vessel for the first time
Build your first business
Complete all basic tutorials
Complete the first tutorial
Receive all trophies
Complete the England campaign
Complete the Netherlands campaign
Complete the France campaign
Complete all 4 campaigns
Build 250 businesses in one game
Build 1,000 businesses in one game
Own at least one business of each kind in a game
Manage 30 cities in one game
Win 50 manual naval battles in one game
Use 8 different tactics in a manual naval battle
Own all types of vessels in one game
Complete 100 town's tasks in one game
Find 10 treasures in one game
Have 50 active trade routes in one game
Destroy 15 pirate hideouts in one game
Plunder 25,000 commodities from trading convoys in one game
Loot a town
Annex 5 towns in one game
Unlock all concessions in one game
Employ 10 captains in one game
Employ 5 captains with maximum level in one game
Sink 1,000 vessels in one game
Employ 10,000 sailors at the same time in one game
Successfully board 100 vessels in one game
Accumulate 1,000,000,000 gold
Manage 20 towns with a satisfaction level of more than 80%
Reach a population of 300,000 in the Caribbean