The Many-Worlds Interpretation of Quantum Mechanics - a brief description for the lay reader, some philosophical considerations, and links to more rigorous treatments   In 1957, Hugh Everett III proposed a radical new way of dealing with some of the more perplexing aspects of quantum mechanics. It became known as the Many-Worlds Interpretation. According to this interpretation, whenever numerous viable possibilities exist, the world splits into many worlds, one world for each different possibility (in this context, the term "worlds" refers to what most people call "universes"). In each of these worlds, everything is identical, except for that one different choice; from that point on, they develop independently, and no communication is possible between them, so the people living in those worlds (and splitting along with them) may have no idea that this is going on.span> In this way, the world branches endlessly. What is "the present" to us, lies in the pasts of an uncountably huge number of different futures. Everything that can happen, does, somewhere. Until Many-Worlds appeared, the generally accepted interpretation of quantum mechanics was (and perhaps still is) the Copenhagen Interpretation. The Copenhagen Interpretation makes a distinction between the observer and the observed; when no one is watching, a system evolves deterministically according to a wave equation, but when someone is watching, the wavefunction of the system "collapses" to the observed state, which is why the act of observing changes the system. The Copenhagen Interpretation gives the observer special status, not accorded to any other object in quantum theory, and cannot explain the observer itself, while Many-Worlds models the entire observer-observee system. The Many-Worlds Interpretation is an interpretation of quantum mechanics, and pertains to quantum events. But it also has implications for macroscopic systems like you and me. Although you may think that there are certain alternatives you would never choose, can you really be sure of that? There are a practically infinite number of versions of you, who have all split off at some time in the past from the path you are now following. There may be versions of you that split off five or ten years ago, or perhaps five minutes after you were born, to whom those choices may not seem unthinkable. But in a very real sense, those people are still "you" (but it can be argued that we should not use the word "are", or even "were"; we need to invent a new kind of tense...) Many people find the Many-Worlds Interpretation, and the consequences that flow from it, deeply disturbing. This includes a great many physicists. It is also apparent that many physicists, including many who teach physics, do not have a good understanding of Many-Worlds. However, polls have been taken among theorists who study such things, and have revealed that most of them believe that the Many-Worlds Interpretation represents, in some sense, an accurate description of the way the world really is. The polls also show that many of them would rather not discuss the subject. It's not hard to see why so many people find these ideas disturbing. For if they are correct, they have profound implications for our understanding of the nature of the Soul, because the Soul (if there is such a thing) must branch along with the worlds that contain it. It would appear that the writings on which many contemporary religions are based make no mention of such an idea. It is commonly thought that Many-Worlds is an unprovable hypothesis, experimentally indistinguishable from the Copenhagen Interpretation, but this may not be the case. It may be possible to observe experimentally one of the predicted effects of Many-Worlds: quantum interference between adjacent worlds. It has even been suggested that the Heisenberg Uncertainty Principle derives from this quantum interference; after you make a measurement (which of course splits the world), you can't be sure about the subsequent state of the observed system, because you can't be sure which world you are in..
This brief description is not very rigorous, in a technical sense, and is intended for the lay reader. Others, far more qualified than I, have written much better on the subject; you can find some of their works by following the links below.
The Everett Interpretation 
Many Worlds FAQ
Tony Smith: Many-Worlds Quantum Theory
Quantum Future Physics
Anders Sandberg: Thoughts and Comments of the Omega Point Theory
Henry P. Stapp:
The Basis Problem in Many-Worlds Theories
Meta-Technology: An Analytical Sketch
James P. Hogan: Paths to Otherwhere Many Worlds
William D. Eshleman: Bill's Many-Worlds Page!
Stephen Paul King: Outlaw Science
Anders Sandberg: Anders Transhuman Page
John G. Cramer: Alternate View Column AV-16
The Anderson/Savage Symposium:
Poul Anderson and Marshall T. Savage
Nick Bostrom:
Observational selection effects and probability
Jürgen Schmidhuber: A Computer Scientist's View of Life, the Universe, and Everything (download the file from here)
Max Tegmark: The Interpretation of Quantum Mechanics: Many Worlds or Many Words?
Max Tegmark: Which mathematical structure is isomorphic to our Universe?
David Deutsch: The Fabric Of Reality
Jacques Mallah:
The Computationalist Wavefunction Interpretation Agenda (CWIA) and
The Observer as a Classical Computer in a Quantum Universe
Don N. Page: Observational Consequences of Many-Worlds Quantum Theory
James Higgo: Quantum Theory of Immortality
Bruno Marchal: Calculabilité, Physique et Cognition
Roy Frieden: Fisher information and the laws of physics Wei Dai: "It may not be true, but it fits on half a page..."
Philip Gibbs: Event-Symmetric Space-Time