Very basically membrane theory is related to string theory. It is the idea of a possible way of explaining how the the 4 forces ( gravity, electromagnetism, strong and weak forces) in the universe might be unified.
One key feature of Membrane theory is related to something called Membrane (brane) cosmology. Our universe basically is part of a 'bulk' of branes (ie an infinite kind of membrane bulk). Gravity is the difficult one to unify into the others. The thought is that gravity is force that 'leaks' through all the bulk and so appears quite weak in our universe.
In this theory it is thought that the Big Bang is the result of two branes touching each other producing the laws of nature and constants we see around us.
An advantage with this approach is that it can be tested with experiments to determine if the strength of gravity for example gets stronger at sub millimetre scales.