Here is the full list of all 14 Animated Puzzles achievements.
Complete all tiers.
Complete the cow puzzle (normal) in stripes, by placcing pieces only on every second row. Begins with top left corner.
Complete tier 1.
Complete a tier puzzle in all difficulties.
Complete all tier puzzles in hard.
Complete all tier puzzles in all difficulties.
Complete all tier puzzles with three different puzzle piece shapes.
Complete the "sunset house" puzzle upside down.
Begin a battle.
Win a battle.
Get a medal in Daily challenges.
Get a diamond medal in Daily challenges.
Complete the dandelion puzzle (normal) like a checker board, by placing only every second piece of the puzzle. Begins with top left corner.
Complete the "black cat" puzzle within 2 minutes. (Normal difficulty)