I hear physicists saying that the "quantum of the gravitational force" is something called a graviton. Doesn't general relativity say that gravity isn't a force at all?
You don't have to accept that gravity is a "force" in order to believe that gravitons might exist. According to QM, anything that behaves like a harmonic oscillator has discrete energy levels, as I said in part 1. General relativity allows gravitational waves, ripples in the geometry of spacetime which travel at the speed of light. Under a certain definition of gravitational energy (a tricky subject), the wave can be said to carry energy. If QM is ever successfully applied to GR, it seems sensible to expect that these oscillations will also possesses discrete "gravitational energies," corresponding to different numbers of gravitons.
Quantum gravity is not yet a complete, established theory, so gravitons are still speculative. It is also unlikely that individual gravitons will be detected any time in the near future.
Furthermore, it is not at all clear that it will be useful to think of gravitational "forces," such as the one that sticks you to the Earth's surface, as mediated by virtual gravitons. The notion of virtual particles mediating static forces comes from perturbation theory, and if there is one thing we know about quantum gravity, it's that the usual way of doing perturbation theory doesn't work.
Quantum field theory is plagued with infinities, which show up in diagrams in which virtual particles go in closed loops. Normally these infinities can be gotten rid of by "renormalization," in which infinite "counterterms" cancel the infinite parts of the diagrams, leaving finite results for experimentally observable quantities. Renormalization works for QED and the other field theories used to describe particle interactions, but it fails when applied to gravity. Graviton loops generate an infinite family of counterterms. The theory ends up with an infinite number of free parameters, and it's no theory at all. Other approaches to quantum gravity are needed, and they might not describe static fields with virtual gravitons.
