An Everett branch is one of the "worlds" in the many-worlds interpretation of quantum mechanics. On a naïve view, the world we perceive around us is all that we have reason to expect exists. However, clever experiments in quantum mechanics tell us that the universe is actually described by a mathematical object called a wavefunction, with our observable universe existing "side-by-side" with many alternatives or "parallel universes". It is difficult to explain the necessary concepts in just a few sentences, and true understanding requires mathematical grounding.

Blog posts

And the Winner is... Many-Worlds! - An index of posts explaining quantum mechanics and the many-worlds interpretation.
Belief in the Implied Invisible - That it's impossible even in principle to observe something sometimes isn't enough to conclude that it doesn't exist.

See also

Many-worlds interpretation, Quantum mechanics
Possible world
Quantum immortality