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

Xipe Imoon Trophy (Bronze): Award for exemplary completion of an Imoon nebula. All levels of the nebula must be completed with a minimum score achieved on each to win this award.
Ymir Imoon Trophy (Bronze): Award for exemplary completion of an Imoon nebula. All levels of the nebula must be completed with a minimum score achieved on each to win this award.
Zeus Imoon Trophy (Bronze): Award for exemplary completion of an Imoon nebula. All levels of the nebula must be completed with a minimum score achieved on each to win this award.
Imoon Agamemnon Trophy (Bronze): This is the award for the successful completion of the first level of the Xipe nebula.
Imoon Technician Trophy (Bronze): This award is for the first successful building of a tower upgraded to level 5.
Imoon-Bane Trophy (Bronze): Awarded for killing a huge number of Insectocytes.
Imoon Survivor (Bronze): Awarded for surviving for an impressive amount of time in any Vengeance Mode level.
Command Degree Trophy (Bronze): Awarded for completing all of the tutorial sections of the game.
Command Credit Trophy (Bronze): Awarded for earning a significant amount of credits during your levels.
Command Skill Credit Trophy (Bronze): Awarded for earning a significant amount of credits during your game using Command Skills.
Imoon Conqueror (Silver): This is the award for completion of all levels of the three Imoon nebulae.
Imoon Artisan (Silver): Award for mastery of technology. This is achieved by researching and building all tower types in the game.
Imoon Admiral (Gold): This is the award for completion of all levels of the three Imoon nebulae without any damage to mining operations.