If this entry is unintelligible, do not be discouraged, because somewhere out there in the multiverse is another version of this entry that is scintillatingly clear. And if you missed NOVA's excellent Parallel Worlds, Parallel Lives episode on Tuesday night, you have through Oct. 28 to watch it online on the PBS website, here. After that you will have to find a parallel universe with a later deadline, or rent the DVD.
The Parallel Worlds program is a brief exploration of the eccentric life and radical work of quantum physicist Hugh Everett, as seen through the eyes of his son, rock musician Mark Everett. The genially shambling Everett the younger is much more interesting here than in his role as "E" of the Eels, an understated and rather minor act. Everett knows zip about mathematics or physics, so he serves as a good host for the uninitiated viewer as the show explores the rudiments of the elder Everett's "many worlds" theory of quantum mechanics -- a theory that is much more popular today, to the point of becoming a pop culture lodestar, than it was in 1957, when it was published.
As a reader and viewer of popularizations of science (generally, the more challenging, the better), I've long been a fan of the idea of parallel universes. I don't think you need to understand the mathematics to appreciate that it seems to answer the fundamental question of why? in a way the rival "Copenhagen interpretation" of quantum mechanics simply does not.
The background Parallel Worlds website is an excellent complement to the program. You can read some original correspondence of Hugh Everett's (I highly recommend these papers; he is an excellent, lucid expositor of his ideas), and even the published version of his original dissertation. I have not tackled that yet, but I think I will try to get through parts of it. There is also a very good, illuminating interview with Hugh Everett's biographer, Peter Byrne. And don't miss the timeline that includes developments related to the many worlds hypothesis in both science and science fiction. There's even a reference to the famous "Mirror, Mirror" episode of the original Star Trek. Although that episode did indeed depict a parallel universe, a more appropriate reference would have been to the "Parallels" episode of Star Trek: TNG, which explicitly dealt with the quantum mechanical theory of infinite parallel universes, and even offered a tidy summary of it (thank you, Data).