A space or structure in M-Theory, the new theory of the nature of the universe. The name comes from membrane, and is a generalisation of a string in string theory. A string is one-dimensional, a membrane is a two-dimensional surface, and a p-brane is a space of any dimension p. So the general term is brane.
M-Theory says that no one way of looking at the universe, like points or strings, is the true way, but higher symmetries become apparent when you move to higher dimensions, and there is no exact number of dimensions that you can't go beyond. (That's my understanding at any rate.)
A new theory of the Big Bang called the ekpyrotic universe model says that the energy release of the Big Bang came about from the collision of two 3-branes in something called bulk space.