A cat is placed in a sealed box. Attached to the box is an apparatus containing a radioactive nucleus and a canister of poison gas. When the nucleus decays, it emits a particle that triggers the apparatus, which opens the canister and kills the cat. According to quantum mechanics, the nucleus is described as a superposition (mixture) of "decayed nucleus" and "undecayed nucleus". However, when the box is opened the experimenter sees only a "decayed nucleus/dead cat" or a "undecayed nucleus/living cat." The question is: when does the system stop existing as a mixture of states and become one or the other? The purpose of the experiment is to illustrate that quantum mechanics is incomplete without some rules to describe when the wave function collapses and the cat becomes dead or alive instead of a mixture of both.
Contrary to popular belief, Schrödinger did not intend this thought experiment to indicate that he believed that the dead-alive cat would actually exist; rather he considered the quantum mechanical theory to be incomplete and not representative of reality in this case.
In the Copenhagen interpretation, a system stops being a mixture of states and becomes one or the other when an observation has taken place. This experiment illustrates the subjective nature of measurement and observation. Some interpret the experiment to mean that while the box is closed, the system simultaneously exists in a mixed superposition of the states "decayed nucleus/dead cat" and "undecayed nucleus/living cat", and that only when the box is opened and an observation performed does the wave function collapse into one of the two states. More intuitively, some feel that the "observation" is taken when a particle from the nucleus hits the detector. However (and this is a key point of the thought experiment), there isn't any rule that says one way or the other, and quantum mechanics is incomplete without such rules and explanations for how such rules come to exist.
In the Everett many-worlds interpretation, which does not single out observation as a special process, both states persist, but decohere. When an observer opens the box, he becomes entangled with the cat, so observer-states corresponding to the cat being alive and dead are formed, and each can have no interaction with the other.
Curiously, all of this has some practical use in quantum cryptography. It is possible to send light that is in a superposition of states down a fiber optic cable. Placing a wiretap in the middle of the cable which intercepts and retransmits the transmission will collapse the wavefunction (in the Copenhagen interpretation, "perform an observation") and cause the light to fall into one state or another. By performing statistical tests on the light received at the other end of the cable, one can tell whether it remains in the superposition of states or has already been observed and retransmitted. In principle, this allows the development of communication systems that cannot be tapped without being noticed at the other end. This experiment (which can be performed, though a workable quantum cryptographic communications system which can transmit large quantities of data has not yet been constructed) also illustrates that "observation" in the Copenhagen interpretation has nothing to do with consciousness, in that a perfectly unconscious wiretap will cause the statistics at the end of the wire to be different.
An interesting variant of the Schrödinger's Cat experiment known as the quantum suicide machine has been proposed by physicist Michael Tegmark. It asks the question, what does the Schrödinger's Cat experiment look like from the point of view of the cat, and argues that this question may be able to distinguish between the Copenhagen interpretation and many worlds.
Another variant on the experiment is Wigner's friend.
Physicist Stephen Hawking is famous for his oft-made statement, "When I hear of Schrödinger's cat, I reach for my gun." This was a deliberately ironic paraphrase of Hermann Goering's anti-intellectual quote, "When I hear the word 'culture', I reach for my revolver", which itself was from a play by German playwright and Nazi Poet Laureate, Hanns Johst[?].
The original article appeared in the german magazine "Naturwissenschaften" ("natural sciences") in 1935:
E. Schrödinger: "Die gegenwärtige Situation in der Quantenmechanik" ("The present situation in quantum mechanics"), Naturwissenschaften, 48, 807, 49, 823, 50, 844 (November 1935)
It was intended as a discussion of the EPR article published by Einstein, Podolsky and Rosen in the same year. Apart from introducing the cat, Schrödinger also coined the term "entanglement" (German: Verschränkung) in his article.
Ursula K. Le Guin wrote a science fiction story called Schrödinger's Cat in 1974. It appeard in The Compass Rose[?] published in 1982. The story deals with Schrödinger's cat, stoves and quantum decoherence. The story is even stranger than Schrödinger's thought experement.
Robert Anton Wilson wrote a trilogy of science fiction/conspiracy theory novels (The Universe Next Door[?], The Trick Top Hat[?], The Homing Pigeons[?]) commonly called the Schrödinger's cat trilogy[?].