Schrödinger's cat is a thought experiment proposed by Dr. Erwin Schrödinger to make a point about quantum mechanics. Wikipedia has more information, so I'll give just a quick rundown.
Dr. Schrödinger's thought experiment was a reductio ad absurdum argument against an interpretation of quantum mechanics. Reductio ad absurdum is Latin for reducing to the absurd, and refers to arguing against a theory, statement, or position by taking it to a ridiculous — but perfectly logical — conclusion. His setup was simple: a cat is placed in a box which also has a mechanism to amplify the activities of subatomic particles in such a way that if a certain event happened, the results would kill the cat. According to quantum mechanics, Dr. Schrödinger said, the cat would be both alive and dead simultaneously while in the box, and you wouldn't know which was actually the case until you looked. (This, of course, did not take into consideration the possibility of undead felines).
This paradox led to some webdevelopers quipping that the only proper use of blinking text was the phrase Schrödinger's cat is not dead.