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