John Stewart Bell was an Irish physicist who outlined an experiment to test the classical assumption that nature works in a "local"--that is, mechanistic--way. The results were to reveal that the classical assumption is wrong--that nature is in some sense nonlocal. From this odd finding sprang considerations so astonishing as to render plausible the physicist Henry Stapp's opinion that bell's theorem constitutes "the most profound discovery in science."
Locality is the supposition that one system can change another only if there is some sort of mechanical interaction between the two. According to relativity, no such interaction can occur at faster-than-light speed. ... To say that fiddling with one particle over here can instantly influence its sister particle over there is to assert that subatomic particles behave in a nonlocal way. This would overthrow the time-honored assumption of locality, and this is what Einstein found so repugnant about the situation, and why he constructed the original EPR (Einstein-Podolsky-Rosen) thought experiment to highlight its apparent irrationality.
... in all experiments conducted since, the verdict is clear: Bohr was right (nonlocal effects do occur in quantum states) and Einstein wrong (there are no hidden variables to explain nonlocality). Nature--on the subatomic scale at least--really is nonlocal. Fiddling with one particle really does mean that its sister particle is altered, instantly, even if it is far away, and neither hidden variables nor any other mechanistic scheme can rescue Einstein's belief in locality.
I note here that this fits rather well into the "Was Einstein wrong?" thread that is currently being debated elsewhere on the board.