Here is the full list of all 20 PUZZLE: BIRDS achievements.
Finish Level 1
Finish Level 2
Finish Level 3
Finish Level 4
Finish Level 5
Finish Level 6
Finish Level 7
Finish Level 8
Finish Level 9
Finish Level 10
Finish Level 11
Finish Level 12
Finish Level 13
Finish Level 14
Finish Level 15
Finish Level 16
Finish Level 17
Finish Level 18
Finish Level 19
Finish Level 20