Here is the full list of all 20 KNIGHTS achievements. It takes around 6-8 hours to unlock all of the achievements on Windows.
Clear a 4x4 Level
Move 400 Pieces
Clear a 4x3 Level
Clear the PAWN Level Pack
Move 40 Pieces
Clear a 3x3 Level
Clear the ROOK Level Pack
Clear the BISHOP Level Pack
Clear the QUEEN Level Pack
Clear the KING Level Pack
Clear a 5x4 Level
Clear a 5x5 Level
Move 4000 Pieces
Unlock Daily Levels
Clear A Daily Level
Clear 5 Daily Levels (Total, Not on the Same Day)
Clear 10 Daily Levels (Total, Not on the Same Day)
Clear 25 Daily Levels (Total, Not on the Same Day)
Clear Level KING-B1 Without Touching Any Yellow Pieces (For Clarification, the KING Level Pack is the Last One)
Obtain All Achievements