Most of the time, people ask this question with the idea of a Newtonian "feather vs. bowling ball" concept in mind. Based on those terms, the typical answer is correct: two objects will fall at the same speed in a vacuum, and air resistance can appear to make an object fall slower. However, there is a surprising, but more complicated nuance to this problem.
Every action has an equal and opposite reaction. This means that, just as the Earth is exerting a gravitational force on the objects, the objects are exerting a gravitational force on the Earth. Just as much as the objects fall onto the Earth, the Earth falls onto the objects as well. It's just the fact that the Earth is so much larger and more massive that we default to viewing things from the first perspective and not the latter. Nevertheless, the gravitational force exerted on the Earth by the objects cannot be ignored.
Gravitational force is determined by the Universal Gravitation law:
`F = (GmM)/r^2`
where m and M are the two masses involved in the interaction. If we do two separate calculations, one for the mass of the lesser object, and one for the mass of the greater object, we can see that there will actually be a larger gravitational force involved with the more massive object.
This is where most people would interject that, well, yes, the larger mass needs a larger force in order to achieve the same acceleration. But reverse the frame of reference; now let's consider this from the point of view of the objects doing the pulling, instead of the Earth. Now we can see that the force exerted by the larger mass is doing more pulling than the smaller mass. The Earth will "hit" the larger mass first.
Mind you, the scale that this takes place on is smaller than we can actually measure with our current technology. However, it makes sense if you consider it in a different way; If you held a brick and the Moon at the same height and dropped them, which would hit first?