Why isn't the acceleration of an object in Earth's gravity related to the object's mass?
This is clearly related to Newton's 2nd law, but since you have asked about the phenomenon rather than the formulas, a Physics SE user explained much better than I can possibly can:


  (..) the gravitational force is different on different masses. Half
  the mass only has half the weight, so it is also only pulled in half
  as much by gravity.
  
  What gravity does is to pull in every single "particle" equally. If
  there are double as many "particles", then the pull in each is still
  the same and each accelerates the same amount.