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

Bail Out (Bronze): Ejected from a damaged aircraft.
Guns Guns Guns (Bronze): Destroyed 5 hostile vehicles from the air as the door gunner.
Fire Hazard (Bronze): Removed enemy presence from the oil field and the base.
Life Saver (Bronze): Rescued Major Illich from enemy captivity.
Fearsome Guardian (Bronze): Protected your allies using the gunship, and led a successful attack on the base.
Emergency (Bronze): Attempted an emergency landing in a damaged aircraft.
Total Annihilation (Bronze): Disintegrated enemy bombers in mid-air, keeping damage to the city to a minimum.
Steel Hunter (Bronze): Eliminated enemy threat from the Suez Canal.
Defender of World Heritage (Bronze): Prevented enemy from using Trinity to destroy Derbent.
Hot Pit (Bronze): Successfully took off to protect allies amid heavy hostile fire.
One Million Tons of Scrap Metal (Bronze): Sank the enemy fleet and saved the Russian Prime Minister.
Nick of Time (Bronze): Destroyed the ICBMs.
Berserker (Bronze): Destroyed the air defense system, opening the way for allied pilots.
Warwolf 1 (Gold): Cleared all missions.
Critical Hit (Silver): Took down a hostile fighter with a charged homing missile.
Successive Kill (Bronze): Took down 3 hostiles in a row with an assault chain.
Hard Strike (Bronze): Wiped out hostile ground forces with ASM.
Eagle Eye (Bronze): Successfully focused on 3 targets.
Stay On Target (Bronze): Obtained and destroyed 10 enemies in a helicopter.
Diverse Strikes (Bronze): Defeated hostiles with all 3 of the gunship's weapons.
Gun Master (Bronze): Took out a large number of hostiles as the door gunner.
Shot Master (Bronze): Took down 5 hostiles with a direct shot.
Pursuit Master (Bronze): Successfully pursued 3 enemies with counter maneuvers.
Switch Master (Silver): Successfully switched from defense to offense 10 times with counter maneuvers.
Checking In (Bronze): Flew through the big hotel's gates.
Limbo (Bronze): Flew beneath a falling chimney.
Speed Demon (Bronze): Destroyed all radar missile defense systems in a short period of time.
Machine Gun Faithful (Bronze): Destroyed an enemy aircraft using only the standard machine gun while piloting an aircraft.
Smooth Flight (Silver): Managed to get Sova 1 to land safely.
Bomber Master (Bronze): Defeated a large number of enemies while flying a bomber.
Chain Master (Silver): Took down 5 hostiles in a row with an assault chain.
Nice Kill (Silver): Shot down a large number of hostile players in competitive online.
Nice Assist (Bronze): Used DFM support or ASM support to join a flight member.
Nice Save (Bronze): Saved an ally being targeted by hostile players in DFM in competitive online.
Veteran Pilot (Silver): Sortied for 50 hours in competitive online or mission co-op.
Warwolf Squadron (Gold): Cleared all levels in mission co-op.
Formation Attack (Silver): Joined the same DFM with two others and shot down a hostile (mission co-op).
The Collector (Bronze): Piloted all aircraft in the game.
Welcome to Mission Co-Op (Bronze): Cleared 1 level in mission co-op.
Aerial Sniper (Bronze): Used DFM to destroy 10 airborne enemies in competitive online.
Ground Pounder (Bronze): Used ASM to destroy 30 ground enemies in competitive online.
Interceptor (Bronze): Used an attack helicopter and destroyed 10 missiles or bombs in competitive online.
World Tour (Silver): Played on all maps in competitive online.
Ace of Aces (Silver): Obtained an A rank for every mission (free mission).
All Rounder (Silver): Obtained all MVPs.

Additionally, there are six secret trophies:

Sierra Hotel (Platinum): Obtained all trophies.
Patriot (Bronze): Protected Moscow from destruction, freeing it from enemy control.
Friendly Fire (Bronze): Alleviated the attack on Miami and helped Guts bail out.
Category 5 (Bronze): Pursued the enemy and shot down Major Illich.
Guardian (Bronze): Protected Washington D.C. from the threat of Trinity, and returned alive.
Calamity (Bronze): Used Trinity to wipe out the enemy (free mission).