Quantum EntanglementIn a situation where you have two particles, A and B, which are connected through quantum entanglement, then the properties of A and B are correlated. For example, the spin of A may be 1/2 and the spin of B may be -1/2, or vice versa. Quantum physics tells us that until a measurement is made, these particles are in a superposition of possible states. The spin of A is both 1/2 and -1/2. (See our article on the Schroedinger's Cat thought experiment for more on this idea. This particular example with particles A and B is a variant of the Einstein-Podolsky-Rosen paradox, often called the EPR Paradox.)
However, once you measure the spin of A, you know for sure the value of B's spin without ever having to measure it directly. (If A has spin 1/2, then B's spin has to be -1/2. If A has spin -1/2, then B's spin has to be 1/2. There are no other alternatives.) The riddle at the heart of Bell's Theorem is how that information gets communicated from particle A to particle B.
Bell's Theorem at WorkJohn Stewart Bell originally proposed the idea for Bell's Theorem in his 1964 paper "On the Einstein Podolsky Rosen paradox." In his analysis, he derived formulas called the Bell inequalities, which are probabilistic statements about how often the spin of particle A and particle B should correlate with each other if normal probability (as opposed to quantum entanglement) were working. These Bell inequalities are violated by quantum physics experiments, which means that one of his basic assumptions had to be false, and there were only two assumptions that fit the bill - either physical reality or locality was failing.
To understand what this means, go back to the experiment described above. You measure particle A's spin. There are two situations that could be the result - either particle B immediately has the opposite spin, or particle B is still in a superposition of states.
If particle B is affected immediately by the measurement of particle A, then this means that the assumption of locality is violated. In other words, somehow a "message" got from particle A to particle B instantaneously, even though they can be separated by a great distance. This would mean that quantum mechanics displays the property of non-locality.
If this instantaneous "message" (i.e., non-locality) doesn't take place, then the only other option is that particle B is still in a superposition of states. The measurement of particle B's spin should therefore be completely independent of the measurement of particle A, and the Bell inequalities represent the percent of the time when the spins of A and B should be correlated in this situation.
Experiments have overwhelmingly shown that the Bell inequalities are violated. The most common interpretation of this result is that the "message" between A and B is instantaneous. (The alternative would be to invalidate the physical reality of B's spin.) Therefore, quantum mechanics seems to display non-locality.
Note: This non-locality in quantum mechanics only relates to the specific information that is entangled between the two particles - the spin in the above example. The measurement of A cannot be used to instantly transmit any sort of other information to B at great distances, and no one observing B will be able to tell independently whether or not A was measured. Under the vast majority of interpretations by respected physicists, this does not allow communication faster than the speed of light.