GR models gravity as a variation of space and time itself: Large bodies and energy densities bend the four-dimensional spacetime in such a way that an attractive effect between bodies is created. In the limit of small energy/mass densities, GR reproduces newtonian-gravity.
Semi-classical gravity refers to the standard model or quantum field theory on a curved spacetime. In other words, gravity is treated as classical whereas everything else is treated a quantum.
Examples of major results from Semi-Classical Gravity include Hawking radiation and the Chandrashekhar limit.
So far, there is no accepted theory of quantum gravity. Similarly to the gauge bosons $\gamma, W^\pm, Z^0$ and the various gluons, which mediate the electromagnetic, weak and strong interactions, another boson, dubbed graviton, is assumed to mediate the gravitational attraction. From the various features of gravity (long-range, always attractive), it is assumed that the graviton is a massless spin-2 boson.
Note that the graviton is not to be confused with the Higgs mechanism, which mediates mass to the gauge bosons in the first place (and has nothing to do with gravity).
One popular approach to quantum gravity is string theory. String theory has been successful in reproducing General Relativity in the low-energy, classical limit. String theory aims not only to be a theory of quantum gravity, but also a theory of everything, which means it also unifies the other gauge forces and matter together. String theory reproduces General Relativity in the non-stringy limit by requiring conformal invariance to constrain the beta-functions to vanish.
String theory requires extra dimensions for conformal anomalies to vanish, and it also requires supersymmetry in order to have fermions in its spectrum. Neither of these have been observed to a conclusive position, though the 125 GeV higgs is a strong evidence for supersymmetry (as in, the mssm, which has been shown to take place in certain realistic string vacua by Kumar, Acharya and Kane) and there has been a recent result hinting at third-generation superpartners being observed at the LHC.
Loop quantum gravity is another well-known theory of quantum gravity that quantises general relativity by using different variables, the Ashtekhar variables instead of the standard spacetime metric (with its corresponding le-cevita, or christoffel connection). Loop Quantum Gravity is formulated as a first-order theory, which means it uses the vielbin (specifically, the vierbin, a vielbin in 4-dimensional spacetime), i.e. the unit vector in curved spacetime. In fact, loop quantum gravity doesn't directly use the vwierbin, but the viewrbin divided by the "Imirizzi parameter".
It is well-known that Loop Quantum Gravity produces a discrete, or granular picture of spacetime; This means LQG does not respect Lorentz invariance, which has been tested to the scale of the planck length. Sen (2013) also showed that Loop Quantum Gravity does not produce a continuous, or smooth picture, of spacetime even at a large scale. There has also been no successful way of reconciling loop quantum gravity with the standard model interactions. Thus, the proponents of Loop Quantum Gravity generally agree that it is unacceptable in its present form.
One related theory is Kaluza - Klein Theory, which attempts to show that General Relativity in a 4 + 1 -dimensional spacetime reduces to general relativity and Maxwell's electromagnetism in a 3 + 1 - dimensional spacetime.
Supergravity is an extension to Kaluza-Klein theory which also covers the Weak and Strong interactions, and incorporates supersymmetry in order to describe fermions as well. Theories of supergravity also arises in the low - energy, classical limit of superstring theories