As he explains at the beginning, he has written a very personal book. In Tegmark's words, this is "a scientific autobiography of sorts: although it's more about physics than it's about me, it's certainly not your standard popular science book … Rather, it's my personal quest for the ultimate nature of reality, which I hope you'll enjoy seeing through my eyes."
Although the theory he wants to argue for is the main focus of the book, he needs to lead up to it by spending some time describing concepts and facts that not all his readers may be familiar with. This he does in the first few chapters, which present the assumptions of modern cosmologists concerning space and time and our place in the universe. He suggests that if you are already familiar with these ideas from your reading of popular science books you may wish to skip most of this introductory material and simply read the summaries he helpfully provides at the end of each chapter. If you are in this "hard-core reader of popular science" category and are eager to reach his own ideas you may choose to follow his advice, but it will probably be worth your while to return later, because Tegmark has an unusual way of explaining the basic facts which is often illuminating and may cast things that you thought you understood quite well in a new light.
Once this groundwork has been covered we come to the main subject of the book— Tegmark's scheme of four "nested" levels of parallel universes. We start from the implications of Alan Guth's theory of inflation, which is widely believed to have operated after the Big Bang to give rise to the universe as we see it today. Inflation leads logically to what Tegmark describes as the Level I multiverse. This, he emphasises, is not a theory but a prediction of inflation; if you accept inflation you must also accept that there are uncountably many universes. These are "universe-sized parts of our space that are so far away that light from them hasn't had time to reach us". This is the multiverse. If it is infinite, as it may well be, infinitely many copies and near-copies of you and everything else will exist in these "pocket universes".
Eternal inflation also predicts the existence of universes with different physical laws. This gives us the Level II multiverse. "Inflation converts potentiality into reality: if the mathematical equations governing uniform space have multiple solutions, then eternal inflation will create infinite regions of space instantiating each of those solutions— this is the Level II multiverse." Tegmark illustrates this idea rather neatly by saying that "students in Level I parallel universes learn the same things in physics class but different things in history class, while students in Level II parallel universes could learn different things in physics class as well."
To understand the Level III multiverse we have to accept the many-worlds interpretation of quantum mechanics put forward by Hugh Everett in 1957. When Everett published his hypothesis it effectively put an end to his career in physics— and Tegmark includes an email he himself received from the editor of a physics journal who warned him that the same fate might befall him! In fact, views are changing today. A number of prominent physicists now advocate Everett's idea; Tegmark regards him as one of the most important scientists of the twentieth century.
The Level III multiverse is a consequence of the many-worlds hypothesis. It is described in Chapter 8 and is probably the most counter-intuitive idea in the book. "This mathematically simplest quantum theory … predicts the existence of parallel universes where you live countless variations of your life." It also predicts that you will not experience the weirdness that this entails, because of a censorship effect within your brain called decoherence. Tegmark finds it impossible to explain all this without introducing mathematical terms such as the Schrödinger wave equation and "the infinite-dimensional place called Hilbert space where it lives". I think that here we are probably reaching the limit of what can be explained non-mathematically.
Finally we come to Tegmark's Level IV multiverse, the Mathematical Universe Hypothesis (MUH). The key idea here is that of a "mathematical structure", which is a "Set of abstract entities with relations between them [which] can be described in a baggage-independent way". By "baggage" Tegmark means "Concepts and words that are invented by us humans for convenience, which aren't necessary for describing the external physical reality". So ideally we should not need concepts such as protons, neutrons, quarks and the rest in order to describe reality.
Tegmark's recipe for understanding reality is to eliminate the "baggage". When he has done this he is left just with the mathematical structure, and this, he believes, is all we need. Mathematical structures are completely abstract, purified from the verbal and conceptual supports that most of us rely on to help us understand difficult concepts. "Mathematical structures are eternal and unchanging: they don't exist in space and time— rather, space and time exist in (some of) them." This is something that Plato would have appreciated.
Among the implications of the MUH are the following.
Both books have a narrow focus in a sense, although the questions they address are anything but narrow. For a wider view of the multiverse idea see The Hidden Reality by Brian Greene; this is one of the books that Tegmark recommends for further reading, although Greene himself is not enthusiastic about the MUH.