Trophies - Successfully complete one of the following tasks to get a trophy:

Enforcer (Bronze): Complete 10 execution styles.
Contract Killer (Bronze): Eliminate 5 made men with the appropriate kill condition.
Three Crime Rings (Bronze): Control three crime rings.
Five Crime Rings (Bronze): Control five crime rings.
Lockpicker (Bronze): Crack five safes.
MobFace (Bronze): Create a mobster using MobFace.
Let Me Upgrade You (Bronze): Upgrade one of your made men's skills or weapons.
Torch The Joint (Bronze): Send your men to bomb a venue from the Don's View.
Pulling The Strings (Bronze): Send your men to attack a venue from the Don's View.
Gun Smuggler (Bronze): Find a level 2 firearm upgrade.
Modified Firepower (Bronze): Find a level 3 firearm upgrade.
Bank Job (Bronze): Heist your first bank.
100 Whacked (Bronze): Kill 100 mobsters.
250 Iced (Bronze): Kill 250 mobsters.
25 Massacred (Bronze): Kill 25 mobsters.
BlackHand Brutality (Bronze): Kill 25 mobsters only using BlackHand attacks.
500 Empty Suits (Bronze): Kill 500 Mobsters.
Full Of Lead (Bronze): Kill 150 mobsters only using firearms.
Fortified Venue (Bronze): Max out all guards at a venue.
Right Hand Man (Bronze): Promote one of your made men to Capo.
Second In Command (Bronze): Promote one of your men to Underboss.
Getting Made (Bronze): Recruit your first soldier.
Shakedown (Silver): Control all extortable businesses.
It's Not Personal (Silver): Eliminate a rival family by successfully bombing their compound.
Bag Man (Silver): Heist each bank once.
750 Sleeping With The Fishes (Silver): Kill 750 mobsters.
This Thing Of Ours (Silver): Recruit a full family tree.
First Crime Ring (Silver): Control one crime ring.
Accept This As A Gift (Silver): Earn one of each favor.
Welcome To The Gun Show (Silver): Collect all level 2 and level 3 firearm upgrades.
Executions In Style (Gold): Complete all execution styles.
Organized Crime (Gold): Control all crime rings.
Safecracker (Gold): Crack all safes.
Platinum Trophy (Platinum): Act like a mobster, think like a Don.

Additionally, there are eight secret trophies:

Vendetta (Bronze): Eliminate three rival families.
Ambush (Bronze): Eliminate four rival families.
Double Crossed (Bronze): Eliminate two rival families.
Paying Tribute (Bronze): Unlock all safehouses.
The Counselor (Bronze): Accept Tom Hagen as your Consigliere.
CIA? DOA (Bronze): Kill Henry Mitchell.
Last Famlily Standing (Bronze): Eliminate all five rival families.
It's Only Business (Bronze): Eliminate Hyman Roth.