Is string theory a quantum theory of gravity? I have read that string theory predicts (or requires ?) the existence of gravitons.
So, would that make it a quantum theory of gravity ?
If so, I have also read that quantum gravity would allow us to understand what happens at the singularity inside a black hole. What does string theory say about this ?
If string theory can't make any predictions about this, what are we missing ? 
Is the theory incomplete ? Or is it that we have multiple string theories - hence, multiple ways to quantize gravity - and don't know which is the right one ?
I'm not a physicist, so just looking for an explanation at the Brian Greene popular book level. :)
 A: String theories are in the forefront of particle theory at the moment because they are the only theories that can quantize gravity and at the same time allow the  incorporation of the Standard Model of particle physics within the theory. The SM can be thought as an efficient mathematical description of all the data gathered up to now for particle physics, there fore any theory of everything should incorporate it in some form.
String theories are a work in progress and in wait for data that will allow determination of constants on what exactly is nature's choice. We might be lucky with the LHC to make progress on this, we might not and will need patience and decades  of experiments and theoretical studies. One can study what happens in black holes using string theory models but the reason they are popular is that they pave the way to a Theory Of Everything. Or so many physicists hope.
So yes, string theories are quantum theories of gravity and include the graviton and a host of other particles that can be counted like angels on the head of a pin. :) . It is data that is needed to solidify a specific string theory model.
A: 
I have read that string theory predicts (or requires ?) the existence of gravitons.

String theory models contain massless spin-2-particles called gravitons, therefore they postdict their existence. The fact that the exchange of such particles results in a GR-like gravitational force was discovered some time before string theory came into being, around 1950, and made popular by Richard Feynman in his lectures on gravitation (see amazon).

So, would that make it a quantum theory of gravity?

Classical wavelike phenomena correspond to quantum mechanical particles ("wave-particle duality"), and since GR describes waves, many people believe that a quantum version should describe gravitons. On the other hand, string theory works with a classical GR background, so that it can describe quantum corrections to a classical situation only. It is an open problem if that is already all of the story, or if a quantum theory of gravity would need to alter the concept of spacetime of classical GR on a more fundamental level.
(To be fair: string theorists know and think about this problem, too, and are working on versions of string theory that incorporate a more fundamental change to spacetime.)

If so, I have also read that quantum gravity would allow us to understand what happens at the singularity inside a black hole. What does string theory say about this?

A basic assumption is this: In a consistent theory of gravitation there should not be any singularities with respect to physically relevant phenomena, and the existence of black hole singularities in GR tell us that classical GR breaks down and should be replaced by a quantum version of gravitation. String theory does not say anything about this, because, today, all formulations of string theory assume the existence of a classical spacetime. A full theory of quantum gravity may be able to explain what happens when a black hole forms in classical GR, for example. Maybe string theory will be able to do so in some future version, but not now (see my previous comment). 

If string theory can't make any predictions about this, what are we missing?

Right know there are more answers to this questions than there are heads on this planet thinking about it. The problem is that string theory starts with an unmotivated and unexplainable assumption about the nature of particles, so there is basically no guidance at all what to do when you run into problems. It's not like that you have a set of axioms derived from a well established theory, run into inconsistencies, and go back to the drawing board to see what axiom may have to be replaced...

Or is it that we have multiple string theories - hence, multiple ways to quantize gravity - and don't know which is the right one?

All string theory variants try to do the quantization of gravity the same way (exchange of gravitons on a classical background), so the existence of a bunch of different "string theories" should not be mixed up the the existence of multiple approaches to the quantization of gravity. The question if it is possible to extract the "right" string theory from the plethora of possible ones is a topic of ongoing debate. As for now, it's basically up to you what you believe.
