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

Bronze Damage (Bronze): Deal 10,000 points cumulative damage to win this award.
Challenges (Bronze): Complete all twenty single player challenges.
Training (Bronze): Complete all the tutorial challenges.
Happy Four Play (Bronze): Play a four player PlayStation Network game.
Get a Lob On (Bronze): Win any PlayStation Network game using just the Grenade.
Silver Damage (Silver): Deal 25,000 points cumulative damage to win this award.
Fast, Pink and Hard Award (Silver): Win a Ranked PlayStation Network game in under 3 minutes.
Gold Damage (Gold): Deal 50,000 points cumulative damage to win this award.

Additionally, there are six secret trophies:

Barrel Buster (Bronze): Shoot 100 barrels.
Body Count (Bronze): Send a cumulative total of 100 worms to meet their maker and win this award.
Greedy Worm (Bronze): Collect a cumulative total of 50 crates and this Trophy is yours.
Hot Foot (Bronze): Detonate a cumulative total of 25 mines and you'll 'trigger' this award.
Shepherd (Bronze): Kill a cumulative total of 15 worms using just the sheep for this award...baaa-gain.
Three Bagger (Bronze): Kill three worms with just one shot to bag this award (includes damage from oil drums and mines).