Here is the full list of all 14 Slide Stories: Bonds & Memories achievements.
Finish the first puzzle in any difficulty
Finish the second puzzle in any difficulty
Finish the third puzzle in any difficulty
Finish the fourth puzzle in any difficulty
Finish the fifth puzzle in any difficulty
Finish the sixth puzzle in any difficulty
Finish the seventh puzzle in any difficulty
Finish all 7 standard puzzles in relaxing mode
Finish all 7 standard puzzles in easy mode
Finish all 7 standard puzzles in fun mode
Finish all 7 standard puzzles in challenging mode
Finish all 7 standard puzzles in pro mode
Finish the secret puzzle in any difficulty
Find the secret puzzle