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

I Am the Just Sword (Bronze): (Arcade/Score Attack) Defeated Hakumen.
Murakumo Activated... (Bronze): (Arcade/Score Attack) Defeated Nu.
Wanderer (Bronze): (Player Match) Played first Player Match.
Voyeur (Bronze): (Player Match) Witnessed over 20 battles.
Ruler of Kagutsuchi (Bronze): (Ranked/Player Match) Fought against all characters.
Be Gentle... It's My First Time. (Bronze): (Ranked Match) Played first Ranked Match.
That Was Incredible! (Bronze): (Ranked Match) Won first victory in a Ranked Match.
You Brute! (Bronze): (Ranked Match) Three consecutive wins in Ranked Matches.
Hands Where I Can See Them (Bronze): (Gallery) Collected over 50 illustrations.
I Like to Watch (Bronze): (Replay Theater) Collected over five replays, other than your own.
Hello World! (Bronze): (Story) Watched the opening.
You're the Best! Around! (Bronze): (Training) Dished out a total of over 1,000,000 damage.
Carpal Tunnel (Bronze): Exceed a total playtime of 25 hours.
Legionnaire (Bronze): Reached Lv.10.
Dante (Silver): (Arcade) Beat Arcade mode on Hell difficulty.
Warrior (Silver): (Player Match) Fought a total of 150 battles.
100 Trials (Silver): (Ranked Match) Fought over 100 battles.
Praetorian Guardsman (Silver): Reached Lv.30.
Leonidas (Gold): (Score Attack) Beat Score Attack mode.
200 Trials (Gold): (Ranked Match) Fought over 200 battles.

Additionally, there are 31 secret trophies:

Cat-a-pult (Bronze): Threw a Kaka kitten using Taokaka's Kitty Litter Special!
You Never Forget Your First (Bronze): Performed your first Barrier Crush.
Devil's Advocate (Bronze): Used Ragna's Gauntlet Hades over 100 times.
Ride the Icening (Bronze): Used Musou Senshouzan or Tosshougeki over 20 times in one round, and finished the opponent off.
Spoonful of Sugar (Bronze): Stayed afloat for over 25 seconds with Rachel.
Irresistible (Bronze): Hit the opponent with Tager's Spark Bolt more than six times in one round.
Hat-Trick (Bronze): In one match, successfully pulled off Thirteen Orphans, The Great Wheel, and All Green.
Beautiful Arakune (Bronze): Had a perfect match using Arakune.
Bangarang! (Bronze): Fought against Bang as Bang, and both activated the Fu-Rin-Ka-Zan.
I Can Rebuild Her... (Bronze): As Carl, decommissioned Nirvana three times in one round.
It's Go Time (Bronze): Performed your first Astral Heat.
Restraining Order (Bronze): Used Barrier Burst over 30 times.
Designated Driver (Bronze): Used Distortion Drive over 100 times.
This Just Got Real (Bronze): Caused a Double Down to occur.
I'm faster than anybody! (Bronze): (Ranked / Player Match) Get First Strike five times in a row.
It's the Only Way to Be Sure (Bronze): Performed a 20 hit combo after the round is over.
Greased Pig (Bronze): Successfully escaped 50 throws.
This Is Important... (Bronze): Heard Noel say, Hands off the panda! over 100 times.
It's over 10,000! (Bronze): Dealt over 10,000 damage using Hakumen, without the use of an Astral Heat.
Activate Termination Protocol (Bronze): Completed a 60+ hit combo using Nu.
Their Numbers Count for Nothing (Bronze): Performed an instant block 300 times.
Stop Hitting Yourself (Bronze): Used Counter Assault 30 times.
Animation Conservation (Bronze): Used Rapid Cancel over 100 times.
In Living Color (Bronze): Used all the color palettes of a single character.
Victory Is an Illusion (Silver): (Arcade / Score Attack) Defeated Unlimited Rachel.
Nothing's Gonna Keep You Down (Silver): Earned a Perfect victory 30 times.
Words Hurt Too (Silver): Let your opponent get a 5,000 hit-point lead on you, then taunt you.
Welcome to the Azure Nightmare (Silver): (Arcade / Score Attack) Defeated Unlimited Ragna, and obtained the power of darkness.
Duh Dun DUNNN! (Silver): (Story) Saw the True Ending.
Discouraging Finish! (Silver): Use an Astral Finish with every character.
Power of the Azure (Platinum Trophy): Conquered the world of BlazBlue.