A Summary of Greene's The Hidden Reality and a link you should check out about repetition.
Greene identifies nine different types of multiverse theory, although there seems to be some overlapping among them. First we encounter the Quilted Universe. If we live in an infinite universe, which is possible, everything, including us and our world, will be duplicated, and not just once but an infinite number of times.
Second, there is the Inflationary Multiverse. The theory of inflation leading to a huge expansion of the universe after the Big Bang allows for the existence of an unimaginably vast number of bubble universes, of which our universe would be one.
Greene's own work has been in string theory, and this generates three more kinds of theory about multiverses. In the Brane Multiverse our universe exists on a three-dimensional 'brane' (short for membrane), which floats on a higher-dimensional expanse potentially populated by other branes - other parallel universes.
In the Cyclic Multiverse, collisions between braneworlds can manifest beginnings similar to the Big Bang, yielding universes that are parallel in time. Alternatively, we can combine inflationary cosmology with string theory to provide for many different bubble universes related to the many different forms that multi-dimenstional string theory can take. This is the Landscape Multiverse.
A different path to the multiverse arises from the 'many worlds' approach to quantum mechanics, which suggests that 'every possibility embodied in its probability waves is realized in one of a vast ensemble of parallel universes'. This gives us the Quantum Multiverse. Probably most people who have read popular accounts of modern physics have encountered this idea, but Greene goes into it in considerable depth and brings out aspects of the theory that are not usually emphasised.
A contemplation of the modern understanding of black holes leads, via information theory and entropy, to the Holographic Multiverse, which asserts that our universe is exactly mirrored by phenomena taking place on a distant bounding surface, a physically equivalent parallel universe. This idea is a distant echo of Plato's extended metaphor of the cave.
The Simulated Multiverse holds that our world, including us, is being simulated in a supercomputer. While this may seem absurd, there seems to be no logical reason why it should not be the case. This, of course, is the theme of the film, The Matrix, which in my view failed to do justice to the richness of the idea on which it was based.
Finally, and most speculatively of all, some physicists have suggested that mathematics is the primary reality, and that every possible universe that is mathematically describable really exists. Greene seems to be rather out of sympathy with this idea, which he calls the Ultimate Multiverse.