Called "Spooky action at a distance" by Albert Einstein, the Bell Theorem is one of the most disturbing consequences of quantum mechanics (even weirder than quantum tunneling and matter waves). The above write-ups describe the theorem excellently, but they omit the details of a quite enlightening experiment by Alain Aspect.
If you excite an atom in a certain way, it will emit a pair of photons in opposite directions. Quantum Theory holds that these two photons will always have perpendicular polarizations (the waves will vibrate at 90 degree angles to each other). Also due to quantum theory, the photons cannot be said to have polarizations until the polarization of one is measured. Therefore, measuring the polarization of one forces the polarization of the other. Previous experiments had shown that the polarization of one photon was indeed always perpendicular to the other.
These results, however, were not very surprising. If you have a bag containing a red marble and a green marble, and you randomly draw out one of the marbles, say, the red one, it isn't at all weird when you open the bag and see that it contains the green marble.
Aspect's experiment was different. In his experiment, he excited the atom, then passed one of the photons through a polarizer which changed the (as yet unmeasured) polarization of one of the photons. When he measured the polarization of each photon, he found that they were still perpendicular. In subsequent experiments, he found that the change was instantaneous regardless of distance. It's as though he drew a marble out of the bag, changed it's color without looking, then opened the bag to find that the other marble had instantly changed color. Spooky indeed.
For the non-faint-of-heart, see
Aspect, A.; Grangier, P.; and Roger, G. "Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities." Phys. Rev. Let. 49, 91-94, 1982.
Aspect, A.; Dalibard, J.; and Roger, G. "Experimental Test of Bell's Inequalities Using Time-Varying Analyzers." Phys. Rev. Let. 49, 1804-1807, 1982.