
  Mass is an object's tendency to resist acceleration.


This applies when both masses you're testing are subjected to identical forces. 

From Newton's Law of Gravity,

$$F = G \frac{M \cdot m}{r^2}$$

It is fairly obvious that the force the Earth exerts on a heavy body is more that what it exerts on a light body, so you can not compare the accelerations by comparing just the masses in this case.

Newton's second law gives $$a = \frac{F}{m} = G \frac{M}{r^2}$$

For two bodies an equal distance away from the Earth's center of mass, you can see that the acceleration for both bodies is indeed, the same and independent of the body mass.

Of course, this assumes there is no drag on the falling body, and that the only appreciable force is that due to Newton's Law.

To summarize: 

There is no difference in the acceleration of two objects of identical shape but different masses. The difference you observe between a feather and a golf ball, for example, is not because of gravity - it's because of the shape and the air resistance the body encounters.

Clarification This answer assumes you're in an inertial reference frame (attached to the center of mass of the falling object and the Earth), and that "falling" means the motion of the object towards this center of mass. 

If you consider falling to mean the motion of the object relative to the Earth, then, because the force exerted on the Earth by a heavier body is more, the Earth will accelerate towards the center of mass ever so slightly, and therefore the object will fall ever so slightly faster.

Why do objects fall at the same acceleration?
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