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