Here is the full list of all 14 Papertris achievements.
Pass the first challenge
Pass all challenges up to level 25
Pass all challenges up to level 50
Pass all challenges up to level 75
Reach the goal from level 1 to level 25
Reach the goal from level 1 to level 50
Reach the goal from level 1 to level 75
Reach the goal from level 1 to level 100
Get a new highscore from 50k points
Get a new highscore from 100k points
Destroy over 50k cubes in total
Destroy over 100k cubes in total
Play a 2 player VS game
Destroy 500 cubes in a game to get another depth