We have all been told, since our earliest school physics lessons, that in the 16th Century Galileo pointed what had always been true, but we had all been stupid to realise: that heavy objects do not fall to the ground faster than lighter ones. But why is this the case? The truth, as you will see, is remarkably elegant while also slightly bizarre.
Everyone knows what Galileo did to prove his theory: he dropped two balls of equal size, but made of different metals so that their weights would be different, from the top of the The Leaning Tower of Pisa, and observed that they hit the ground at the same time. This may be an apocryphal story, but it is how everyone learns the fundamental non-intuitive fact of mechanics: that everything falls to the ground at the same speed, regardless of its mass. The only thing that can make a difference is air resistance, and that depends on the shape of the object falling, not its weight.
The pedants among you will point out that it is actually the acceleration that is the same for all falling bodies, not the speed; I have simply said "speed" because we are dealing here with objects that are released at the same time so that at every point their speed will be the same.
Now that is all well and good, but it is still rather strange. Not only is it counter-intuitive, but the fact of equal acceleration seems to contradict the basic idea of gravity: that everything with mass attracts everything else, with a force of attraction somehow proportional to the combined mass of the objects and the distance between them. So a heavier object should surely attract, and be attracted to the Earth more strongly, hence the force of gravity will be stronger, and it will accelerate faster than any similarly-sized, but lighter objects.
To answer this, consider how we might measure an object's mass. To do so, we need to find an interaction between two objects that is dependant on their masses. For example, shooting a snooker ball across a table into a second ball. We measure the speed of the first ball, and when it impacts the second ball, if it stops dead in its tracks while the other ball is knocked away at precisely the same speed, we can pronounce the two balls to have an equal mass. What have we done here? We have taken the mass of the first ball as a reference -- something to compare other masses to, by giving their mass as a multiple of the reference mass -- and we have discovered the mass of the second ball by observing their relative speeds when subjected to the same amount of momentum.
How else might we measure mass? Well, a way that is perhaps more obvious than the first method is to weigh the object. Let us take a set of balance scales -- the old fashioned type, as seen in hieroglyphics, 19th Century kitchens, imagery of justice, my kitchen &c. We place the first snooker ball on one side of the scales -- this is the reference mass. The second ball is placed on the other side, and we report its mass as a multiple of the reference mass -- hopefully this ball will precisely balance the first ball, and we can say that the two masses are equal. What have we done here? We have measured the relative attraction of two objects to the Earth -- pulled by its gravity towards the centre of the planet.
So we have two ways of measuring mass. Well, why does that surprise us? For example, we can measure, say, voltage, with a variety of different analogue or digital devices -- temperature with either an alcohol or a mercury thermometer -- distance with a metre ruler or a trundle wheel. Why should it surprise us that mass can be measured in two different ways? The difference here is that these two ways are fundamentally different. In all those other examples, each different way of measuring is still the same phenomena in fundamentally the same way. However, we have shown that an object's mass exerts an influence over two entirely separate phenomena -- both the object's gravitational attraction and its propensity to oppose a change in velocity (momentum, or inertia), are affected by its mass.
The thing is, as far as physicists can tell, all of this happens as by magic: there is no particularly compelling reason why the mass that an object "uses" in its gravitational interactions should be the same as the mass it "uses" when it is called upon to collide with other objects and do things involving momentum. It is merely an observational fact that when we look at an object's gravitational mass and its inertial mass, we find them to be the same(1).
So, bringing this back to the original question -- when objects fall under the influence of gravity, we find that this accidental equivalence of the two types of mass results in the following happening: a more gravitationally massive object will be attracted to the earth more strongly -- so its downwards acceleration would be stronger, were it not for its increased inertial mass precisely cancelling out this increased acceleration (with more inertial mass, the object has more momentum, so a larger force is required to accelerate it in the first place).
And as we know that gravitational and inertial mass just happen to be the same, this cancelling out is no shock to us.
There is one important caveat, however(2): the more massive an object is (whatever its shape), the less it is affected by air resistance (it has more kinetic energy and momentum). This part is slightly intuitive -- falling heavy objects do not get blown away in windy weather to the same degree as lighter ones.
So why are inertial and gravitational mass the same? No-one knows for sure. It must be said that Einstein's relativity does complicate matters somewhat, for reasons that are beyond the scope of this writeup, but in some sense the unexplained coincidence remains.
In fact, in a flight of fancy, we may ask why inertial mass should not be related to some other arbitrary property of matter. For example, charge is in some ways similar to gravitational mass -- charge affects an object's attraction-at-a-distance to other objects with the electromagnetic force, via an apparently similar mechanism to how gravitational mass affects the object's attraction-at-a-distance via the force of gravity. So, why not have inertial mass proportional to charge rather than gravitational mass?(3)
There is, of course, active development on theories explaining all this (the general idea that gravitational and inertial mass are the same is related to Einstein's Equivalence Principle), but unfortunately nothing conclusive yet.
(1) Technically all we see is that relationship of the two "types" of mass is linear and constant -- we simply make them equal because we are good at choosing our units.
(2) Thanks to soren015 pointing this out. I was too caught up in assuming zero air resistance!
(3) Yes, the electromagnetic force is quite different from gravity, because it can be negative and positive. This was just an example of an arbitrary "property" that an object could have. How about its angular momentum, or even something more obscure and quantum mechanics-related, such as charm?