Here is the full list of all 22 Unium achievements. It takes around 1-2 hours to unlock all of the achievements on Windows.
Solve 30 Medium puzzles.
Solve 20 Medium puzzles
Solve "Spectacles" without using either of the neutral tiles.
Solve 10 Medium puzzles
Solve "Hot Coffee" without using the neutral tile.
Solve "Helmet" by overlapping the line twice.
Solve "Imbalanced Load" by overlapping the line three times.
Solve "Top Hat" by overlapping the line twice.
Solve 10 Easy puzzles
Solve "Fish" by overlapping the fish's eye.
Solve "Lorry" by overlapping the window.
Solve "Battery" by overlapping the line twice.
Solve "Mushroom" by overlapping the line twice.
Solve 10 Beginner puzzles
Solve 10 Advanced puzzles
Solve 20 Advanced puzzles
Solve 30 Advanced puzzles
Solve 40 Advanced puzzles
Solve 50 Advanced puzzles
Solve "The Only Card You Need" two ways: Once crossing the line on the center neutral tile, and once without.
Make a moth out of "Flutterby" by turning all tiles white, except for 3,4 and 8,4.
Solve "Punnett" by overlapping all four tiles in all three white squares.