http://en.wikipedia.org/wiki/Loop_quantum_gravity

The deep origin of the problem is the fact that gravity is geometry. When the quantum properties of gravity are not disregarded, spacetime itself becomes a quantum object, and therefore the usual logic of conventional quantum field theory, which requires the existence of a well defined classical geometry, does not work anymore. This is the starting point of LQG. A quantum theory defined without assuming a classical spacetime is called background independent. LQG is essentially a technique for studying quantum field theory when there is no spacetime background. The quantum states of the theory, labelled by spin networks, should not be thought as living inside a physical space, but rather to define physical space themselves. This is the quantum version of the main property of Einstein's general relativity, where the solutions of the theory are not gravitational field living inside a spacetime, but are themselves defining spacetime.

The theory of LQG is one of the possible solutions of the problem of quantum gravity, along with string theory. There are substantial differences, however, with string theory. String theory addresses also another major open problem in fundamental physics besides quantum gravity: the problem of unification, namely understanding all known forces and particles as manifestations of a single entity. To this aim, string theory postulates the existence of extra dimensions and so-far unobserved particles and symmetries. LQG, on the contrary, is based only on quantum theory and general relativity and its scope is limited to understanding the quantum aspects of the gravitational interaction. On the other hand, the consequences of LQG are radical, because they change in depth our understanding of the nature of space and time and provide a tentative but detailed physical and mathematical picture of quantum spacetime.

my very rough take on that, and the linked paper, is that string theory is a huge artifice, that tries to cover everything, "starting" in a sense, with "generalising" quantum mechanics into something very abstract that can then be used to "pull in" relativity. in contrast, loop theory goes back to the geometric basics of relativity and tries to reconstruct those in a way that would be consistent with quantum mechanics, in the hope that rebuilding relativity on that will allow quantum mechanics to be pulled in naturally.

so they have very different styles. string theory is trying to do everything, and is getting top-heavy. loop theory is a "back to basics" that is trying to re-do relativity in the hope that qm can then be connected back on (perhaps still with pieces of string theory, to make the qm more elegant).

i have no idea what the current take on all this is, though - the paper linked here seems to be from back in 2003.
https://www.google.com/trends/explore#q=string%20theory%2C%2... - it looks like loop theory may have stalled?

evidence of current interest - http://phys.org/news/2012-01-physicists-loop-quantum-gravity... (sounds like the kind of thing that atomic optics or whatever it is called could help with - i think there was a link here recently showing how those might improve gravitational wave detection).