I have to admit that I had never heard of quantum immortality until I read Robert Charles Wilson's short story, "Divided by Infinity." (That's the same Wilson who wrote Spin, which I recommended in a previous post). You can read "Divided by Infinity" online at this site. Take a look at it before I spoil the story for you in the rest of my post.
In Wilson's story, the main character, despondent after the death of his wife, keeps getting closer and closer to committing suicide. But he never actually kills himself. Instead, he finds himself driven into ever more unlikely and bizarre circumstances. Finally, he ends up as the sole survivor of a gamma-ray burst, resurrected by aliens from scraps of his DNA. What is going on here?
The story is based on the concept of quantum immortality, which claims that the many-worlds interpretation of quantum mechanics is correct, and, furthermore, that you can only find yourself in a branch of reality in which you are still alive. Ergo, the universe will always conspire to keep you alive, no matter how unlikely that seems. (Don't think too much about all of your acquaintances who have died -- the immortality guarantee only applies to you in the universe you find yourself in).
Wilson's story is the one most expressly based on this idea (and the story itself gives an excellent explanation of quantum immortality), but there's also a hint of it in Greg Egan's novel, Quarantine. It appears that the idea itself grew out of an attempt to try to come up with some sort of experimental consequence of the many-worlds idea. And like all good scientific theories, it can be verified or refuted experimentally. Just wait around for the next 90 years and see whether or not you die.