Inertial and gravitational mass

An interesting quandary in general relativity is the equivalence of inertial and gravitational mass. These are two theoretically distinct concepts that turn out to be entirely identical. There are two ways of defining mass: according to its interaction with gravity, and according to the force necessary to accelerate it.
Inertial mass is the mass acted on in $\vec F=m_i\vec a$. One way to look at this is that if you have a relatively frictionless table, it takes a lot more force to get a very massive object moving than a lighter one, independent of the force of gravity.
Gravitational mass is the thing on which gravity acts (umm...duh?). The magnitude of gravitational force between two objects of mass $M$ and $m_g$ is $G\frac{Mm_g}{r^2}$. In this way, gravitational mass is the equivalent of charge for the electromagnetic interactions; more massive objects feel more gravitational force.
Intuitively, there is no reason for the gravitational mass $m_g$ and the inertial mass $m_i$ of an object to be the same. But as we all know, they are. This gives us convenient results such as an acceleration due to gravity that doesn't care about the mass of the object.
In general relativity, this equivalence is explained by the equivalence principle, which states that acceleration and gravity are locally indistinguishable.
One of the tests of general relativity is therefore checking whether these two masses are really equivalent. Tests of the equivalence principle have been carried out for a long while, starting with Galileo dropping objects of different masses off the leaning tower of Pisa to check that they accelerated in the same way. More recently, experiments have checked this equivalence more precisely, typically using torsional springs to examine the acceleration of various objects due to gravity.