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

Theorist (Bronze): Complete all Tutorials.
Lieutenant (Bronze): Complete 25% of Campaign mode.
Colonel (Bronze): Complete 50% of Campaign mode.
General (Bronze): Complete 75% of Campaign mode.
Extirpator (Bronze): Complete Campaign mode in 60 minutes or less.
Vanquisher (Bronze): Complete Campaign mode.
Gatherer (Bronze): Gain 9 different rewards in Campaign mode.
Amateur (Bronze): Complete Skirmish on Normal or higher difficulty level.
Comrade (Bronze): Play 100 Local Multiplayer Matches.
Napoleon (Silver): Complete Campaign Mode on Hard difficulty level.
Collector (Silver): Gain 200 rewards in Campaign mode.
Ace (Silver): Complete Skirmish on Hard difficulty level.

Additionally, there are six secret trophies:

Torch-bearer (Bronze): Keep 5th morale level for 25 seconds in Single Player mode.
Escapee (Bronze): In Single Player capture enemy's village after all own buildings were lost.
Midas (Bronze): Upgrade the village to the 5th level in Single Player mode.
Encourager (Bronze): Send a surge containing 1000 units in Single Player mode.
Aggressor (Bronze): Capture 2000 buildings in Single Player mode.
Blighter (Bronze): Kill your 32000 units in Single Player mode.