The visible, three-dimensional universe is restricted to a brane inside a higher-dimensional space, called the "bulk". If the additional dimensions are known, then the observed universe contains the extra dimension, and the bulk is not necessary. With the bulk model, there may be infinite extra dimensions, and other branes may be moving through this bulk. Interactions with the bulk, and possibly with other branes, can influence our brane.
A p-dimensional brane is called a "p-brane": a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one, etc.
D-branes are classified by their spatial dimension, which is one less than their p-dimension. A D0-brane is a single point, a D1-brane is a line, a D2-brane is a plane, and a D25-brane fills the highest-dimensional space in string theory.
Branes are dynamical objects which can propagate through spacetime according to the laws of physics.
The LED model postulates that our 4d universe exists on a brane in a higher dimensional space. It is then suggested that the other forces of nature (the electromagnetic force, strong interaction, and weak interaction) operate within this membrane and its four dimensions, while the gravitational particle graviton can propagate across the extra dimensions. This would explain why gravity is very weak compared to the other fundamental forces.
A brane multiverse postulates that a universe exists on a membrane (brane) which floats in a higher dimension. In this bulk, there are other membranes with their own universes. These universes can interact with one another, and when they collide, the violence and energy produced is more than enough to give rise to a Big Bang. The branes float or drift near each other in the bulk, and every few trillion years, attracted by gravity or other forces, collide and bang into each other. This repeated contact gives rise to multiple or "cyclic" big bangs. This requires access to 6d.