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