Brane cosmology is a version of string theory. The idea is to solve problems in cosmology using speculative particle physics theories and in turn use cosmological observations to motivate ideas in string theory.
The central idea is that our visible, four-dimensional universe is entirely restricted to a brane in a higher-dimensional space, called the bulk. The additional dimensions may be compact. Other branes may be moving through this bulk. Interactions with the bulk, and possibly with other branes, can influence our brane and thus introduce effects not seen in more standard cosmological models.
As one of its attractive features, the model can "explain" the weakness of gravity relative to the other fundamental forces of nature. In the brane picture, the other three forces (electromagnetism and the weak and strong nuclear forces) are localised on the brane, but gravity has no such constraint and so much of its attractive power "leaks" into the bulk. As a consequence, the force of gravity should appear significantly stronger on small (sub-millimetre) scales, where less gravitational force has "leaked". Various experiments are currently underway to test this.