Quantum Gravity, is a theory which attempts to join the theories of quantum mechanics, and general relativity. The purpose of this theory is the creation of a single theory which outlines all fundamental forces. The benefits of this would be vast, as we would have a complete and unified theory to describe fundamental forces. The difficulty of creating such a theory is the fact that the foundation of the two two theories being united are polar opposites, making the situation far more complex.
Quantum mechanics describes three of the four fundamental forces of nature. The first of the three is strong interaction, a strong but short ranged force acting on 10^-13 cm distances, the main use of this force is that it effectively holds the nuclei of an atom together. The second force is known as electromagnetic force, which exists only between pieces of matter carrying an electrical charge; this is the force which causes magnetic effects. The third and final force is known as weak force, which is not only weak, but short ranged, causing both neutrino interactions and radioactive decay.
General relativity describes the fourth fundamental force of nature, gravitational force. This force is weak, similar to the weak force; however, unlike the weak force, it is extremely long ranged,. This forces strength depends on the masses of the two pieces of matter being attracted.
Now that you know what both theories are created to do, you can see that the combination of all four fundamental forces would be a giant leap in the right direction for physics, however there is a significant, yet highly technical difference between the two. General relativity models gravity as a curvature within space-time that changes as mass moves. This is due to the principle in general relativity that there is no fixed space-time background. However, quantum mechanics, in most theories, depends on particle fields embedded in the flat space-time of special relativity, which is to say that there is a fixed space-time background. The practical reality of these two very different statements is that if we apply quantum physics to gravity, and discount the general relativity model, the gravitational particles will group together infinitely, making it impossible to currently compute.
There are many different theories of quantum gravity, each extremely different, yet equally complex. Some of these theories produce a unified field theory, creating a theory of everything. However, some theories, such as loop quantum gravity simply explain gravity using the same foundations developed in quantum mechanics. Some of these include: