Here is the full list of all 20 Downwell achievements.
Beat the boss
Beat the first area (hard mode)
Beat the second area (hard mode)
Beat the third area (hard mode)
Beat the fourth area (hard mode)
Land a 100 combo
Have more than 3000 gems
Beat the fourth area
Complete a level without landing
Land a 30 combo
Land a 10 combo
Beat the third area
Beat the second area
Complete a level without shooting
Complete a level without taking damage
Complete a level without visiting siderooms
Beat the first area
Beat the boss (hard mode)
Complete a level without killing any enemy
Have more than 5000 gems