The central idea is that the visible, four-dimensional universe is restricted to a brane inside a higher-dimensional space, called the "bulk". The additional dimensions are compact, in which case the observed universe contains the extra dimensions, and then no reference to the bulk is appropriate in this context. In the bulk model, 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, thus solving the so-called hierarchy problem. 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 (subatomic or at least sub-millimetre) scales, where less gravitational force has "leaked". Various experiments are currently underway to test this.
The Randall-Sundrum (RS1 and RS2; see 5 dimensional warped geometry theory for a nontechnical explanation of RS1), pre-big bang, ekpyrotic and cyclic scenarios are particular models of brane cosmology which have attracted a considerable amount of attention. The ekpyrotic theory hypothesizes that the origin of the observable universe occurred when two parallel branes collided.