Brane cosmology refers to several theories in particle physics and cosmology motivated by, but not exclusively derived from, superstring theory and M-theory.
The central idea is that the visible, four-dimensional universe is restricted to a brane inside a higher-dimensional space, called the "bulk". If the additional dimensions are compact, then the observed universe contains the extra dimensions, and then no reference to the bulk is appropriate. In the bulk model, at least some of the extra dimensions are extensive (possibly infinite), and 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.
The model can explain the weakness of gravity relative to the other fundamental forces of nature, thus solving the so-called hierarchy problem. In the brane picture, the other three forces (electromagnetism and the weak and strong nuclear forces) are localized 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 (subatomic or at least sub-millimetre) scales, where less gravitational force has "leaked". Various experiments are currently underway to test this.