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

Scrap Metal (Bronze): Destroy 27 Stalkers.
Totaled Eclipse (Bronze): Destroy 9 Eclipses.
Qualify for Duty (Bronze): Complete the Tutorial.
Clean Sweep (Bronze): Destroy all hiding Sai insurgents in the Tutorial.
Ulterior Motives (Bronze): Use the playbook to switch weapons.
First Assault (Bronze): Complete Mission 1: Domestic Disturbance.
In Transit (Bronze): Secure the Control Node powering the turrets in Mission 2: Assault Charge.
Telepathic (Bronze): Establish a portal in Mission 3: Double Jeopardy.
H.Y.D.R.A. (Bronze): Defeat the Hydra in Mission 4: Multiple Counts.
Hammer Time (Bronze): Take out the Arc-Hammers in Mission 5: Self Defense.
Evacuation (Bronze): Rescue Vantage in Mission 6: Innocent Victim.
A Bridge Not Too Far (Bronze): Secure all 3 bridges in Mission 7: Unfair Surprise.
Power Up (Bronze): Secure all 3 power junctions in Mission 8: Reasonable Doubt.
Fur Coat (Bronze): Defeat Sable in Mission 9: Shock Verdict.
Rite of Passage (Bronze): Rescue Eona in Mission 10: Prison Break.
In Pursuit (Bronze): Secure the AA Turrets in Mission 11: Hot Pursuit.
Reverse Polarity (Bronze): Recalibrate the airfield station in Mission 12: Natural Justice.
Spray and Pray (Bronze): Use the Matriarch's Acid Rain ability.
Demolisher (Bronze): Destroy 55 Control Nodes with the Demo Bomb ability.
Puppet Master (Bronze): Destroy an enemy unit with one you have Mind-Controlled.
Unleashed (Bronze): Use the Rage Smash ability.
Playing Both Sides (Bronze): Win 8 ranked matches with each faction.
Demoraliser (Bronze): Destroy 11 enemy units in the first 5 minutes of a ranked match.
First Down and Ten (Bronze): Win a ranked match in under 10 minutes.
Feeding the Habit (Bronze): Play 16 ranked matches.
Calling all Units (Bronze): Recruit one of every unit type in any game mode.
Gone Shopping (Bronze): Lose a multiplayer match while the deployment queue is open.
Maximum Firepower (Bronze): Upgrade a Control Node to have level 3 turrets (multiplayer or skirmish).
Slice and Dice (Bronze): Use the Spectre's Rapid Slice ability to kill an enemy commander (multiplayer or skirmish).
False Sense of Security (Bronze): Attack 24 cloaked Spectres while using the Infiltrator's Thermal Vision.
Appetite for Destruction (Bronze): Destroy 77 units using the Stalker's SAMs.
Pest Control (Bronze): Kill 80 Broodlings.
Veteran's Affair (Silver): Win 99 ranked matches.
Eradicator (Silver): Beat the AI on all Skirmish maps.
Epic Campaign (Silver): Complete Story Mode on Easy or Normal.
The Other Side (Silver): Complete Act 1 of Story Mode.
The Other Path (Silver): Complete Act 2 of Story Mode.
Winning Streak (Silver): Win 6 consecutive ranked matches.
Epic Saga (Gold): Complete Story Mode on Hard.
Full Repertoire (Gold): Use all unit abilities at least once.
Platinum (Platinum): This is the platinum trophy and is unlocked automatically when all the trophies have been unlocked.

Additionally, there are three secret trophies:

Whip it Good (Bronze): Whip 250 times during any map or match.
Decked Out (Bronze): Win 10 ranked matches fully equipped.
Formidable (Bronze): Destroy 500 units.