Parallel universes are basically the most popular trope in science fiction, because Batman and Spider-Man in alternate universe costumes are awesome, and also human life is inherently full of awful, soul-crushing regret. But how does the science actually work?
Our old friend MinutePhysics explains that there are three models of multiverse theory. There's the Bubble Universe/Black hole theory, which states that our observable universe might just be one part of a greater whole, and other parts of the universe are so far away we'll never see them, or are inside black holes, and therefore unobservable. Then there's String Theory's take, that we exist on a three dimensional plane inside of a larger nine-dimensional universe, wherein other three dimensional "membranes" (universes) could exist. And finally there's the quantum mechanical one you're probably most familiar with, where every multiple choice problem that's ever happened in the world (will you step in that puddle? Will you NOT step in that puddle and fall into a sewer and marry a family of rat kings instead?) actually cause new universes to form, with each possibility actually happening in one.
The thing to remember is that these aren't just philosophical quandaries. None of the models have been proven, but tests exist, or are being developed, to test and attempt to confirm them. [MinutePhysics]