If we were able to look at a piece of space the size of the Planck length, the vacuum fluctuations cause that space to become "frothy", or become what is known as "quantum foam". Space ceases to be smooth and continuous, and becomes probabilistic, in a quantum mechanical sense; essentially an extension of the Uncertainty Principle. Superstring theories indicate that at these scales space cannot be described by standard geometries.
Loop quantum gravity starts from the premise that the Planck length is the quantum of space; that is, it is the smallest meaningful distance. In classical physics, if one takes a line, it can be cut in half, then cut in half again, and again continuously. LQG tells us that once the cut is at the Planck length, one can go not further in much the same way as the quantum of light is the photon which is indivisible. The implication is that volumes in space are multiples of a sort of "Planck volume" corresponding to approximately 10-99 cubic cms.
LQG uses "spin networks", called spin foams, that similar to those developed by Roger Penrose and discussed in the section on Twistors & Spinors. Spin foams do not exist within spacetime, they are spacetime, so topology and dimensionality are inherent characteristics of the spin foam. One advantage of LQG over superstring theories is that it is background independent, where Superstrings require a pre-defined space-time. On the other hand, LQG does not adequately describe interactions between gravitons and other fundamental particles. Although General Relativity was a starting point for LQG, it has not been shown to reduce to general relativity at the limit, which is essential for it to concur with the experimentally proven aspects of General Relativity.
LQG does make one very interesting prediction. The constancy of the speed of light may cease to apply to photons at extremely high energy levels which would travel very, very slightly faster than lower energy photons. This is under investigation in the context of very high energy gamma rays arriving from gamma ray bursts occurring billions of light years away. The time differences could be of the order of a few seconds in this scenario.
Two major differences kept LQG and string theory apart. LQG requires neither the higher dimensions that apply to string and M theories, or Supersymmetry. Recently a group under Thomas Thiemann in Germany extended LQG to higher dimensions, and have been able to include supersymmetry. However, string theorists continue to have doubts as LQG does not respect the symmetries of special relativity.
Here is a New Scientist article on Loop Quantum Gravity, and here is a more in depth article on Loop Quantum Gravity; this is fairly mathematical.
Loop Quantum Gravity