Quantum suicide refers to the following thought experiment: I build a suicide device, such as a gun which will fire a bullet into my head, which can be triggered automatically by some external triggering mechanism. I then connect that triggering mechanism to some source of quantum indeterminancy, such as a radioactive sample. This is essentially Schrödinger's cat, with myself as a cat.
What makes this interesting, is that if the many-worlds interpretation of quantum mechanics is true, then at the point at which a decay might happen, the universe splits in two — into one universe in which it decays and I die, and another in which it does not decay and I live. Assuming there is no afterlife, I will cease to exist in one universe but not in another. So, the argument goes, although there will be others who will exist in the universes in which I die, I will only ever exist in the universes in which I survive, so I will only ever observe the universes in which I survive. From my perspective, I will never die, I will always be saved from death by quantum indeterminancy.
Connected to this is the theory of quantum immortality, which posits that no one ever dies, they only appear to. Whenever I might die, there will be another universe in which I still live, some quantum event (however remotely unlikely) which saves me from death. Hence, it is argued, I will never actually experience my own death, but from my own perspective will live forever, even as countless others will witness me die countless times. Life however will get very lonely, since everyone I know will eventually die (from my perspective), and it will seem I am the only one who is living forever — in fact, everyone else is living forever also, but in different universes from me.
However, one could imagine situations in which this doesn't work. If you are 1 millisecond away from having a nuclear weapon explode right above your head, for example. Is a random quantum event really going to recreate a surviving copy of you somewhere else in the world, sufficiently far away from the explosion? Can the quantum vacuum really bubble up fully-grown human beings out of thin air?
Generally speaking, this is not a serious theory of immortality — rather, it is an attempt to explore some of the difficulties posed by the many-worlds interpretation. It can even be viewed as an attempt at a reductio ad absurdum against many-worlds. For obvious reasons, it is neither verifiable nor falsifiable.