Quantum immortality is a metaphysical speculation derived from the quantum suicide thought experiment. It states that the many-worlds interpretation of quantum mechanics implies that conscious beings are immortal.
A physicist sits in front of a gun which is triggered or not triggered depending on the decay of some radioactive atom. With each run of the experiment there is a 50-50 chance that the gun will be triggered and the physicist will die. If the Copenhagen interpretation is correct, then the gun will eventually be triggered and the physicist will die. If the many-worlds interpretation is correct then at each run of the experiment the physicist will be split into one world in which he lives and another world in which he dies. After many runs of the experiment, there will be many worlds. In the worlds where the physicist dies, he will cease to exist.
However, from the point of view of the non-dead copies of the physicist, the experiment will continue running without his ceasing to exist, because at each branch, he will only be able to observe the result in the world in which he survives, and if many-worlds is correct, the surviving copies of the physicist will notice that he never seems to die, therefore "proving" himself to be immortal, at least from his own point of view.
Another example is where a physicist detonates a nuclear bomb beside himself. In almost all parallel universes, the nuclear explosion will vaporize the physicist. However, there should be a small set of alternative universes in which the physicist somehow survives (i.e. the set of universes which support a "miraculous" survival scenario). The idea behind quantum immortality is that the physicist will remain alive in, and thus remain able to experience, at least one of the universes in this set, even though these universes form a tiny subset of all possible universes. Over time the physicist would therefore never perceive his or her own death.