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

Ring (Bronze): Complete 'Ring' Level.
Goblet (Bronze): Complete 'Goblet' Level.
Rod (Bronze): Complete 'Rod' Level.
Sword (Bronze): Complete 'Sword' Level.
Shield (Bronze): Complete 'Shield' Level.
Crown (Bronze): Complete 'Crown' Level.
Bronze Master (Bronze): Collect 20 bronze medals in Expert or 30 in Beginner mode.
Silver Master (Bronze): Collect 15 silver medals in Expert or 20 in Beginner mode.
Gold Master (Bronze): Collect 10 gold medals in Expert or 15 in Beginner mode.
Speed Master (Bronze): Complete 20 levels with lightning speed.
Equilibrist (Silver): Complete 15 levels in a row without falling.
Beginner Master (Silver): Complete All Beginner levels.
Expert Master (Silver): Complete All Expert levels.

Additionally, there are three secret trophies:

Ground Breaker (Bronze): Began playing Cuboid.
Abyss (Bronze): 50 falls.
Determined (Bronze): Spent 8 hours playing Cuboid.