Fifty years ago this month, John S. Bell submitted an elegant little paper to an obscure journal that proved Albert Einstein wrong on an important question. The paper itself is so wonderful that I've linked to the original below.
Quantum mechanics says that in a certain kind of experiment, two particles might be emitted that travel off at high speed in opposite directions, and will have an entangled property -- regardless of the direction in which you measure (and you an pick an arbitrary direction, even long after the particles are emitted), one particle will be measured as having opposite spin along that direction as the other.
Quantum mechanics also says there is no way whatsoever to predict in advance which will will be which (that is, say, which will be spin up and which spin down if you measure along that axis). Note that you can pick any axis to measure along, and instantly, the other particle will be seen to have a correlated spin along that axis, no matter how far away the other particle might now be! (You can't communicate information this way, so it doesn't violate relativity, but it still is, in Einstein's words, "spooky action at a distance".)
Einstein, who hated quantum mechanics even though he was one of its creators, said this was ridiculous, that clearly there is some sort of underlying information about the system that we just don't know -- quantum mechanics must be an incomplete description of the world, and if only we had a complete theory that provided us with all the relevant "hidden variables" that describe the state of the particles, we would know exactly what we would measure in advance, and there would be no "spooky action at a distance".
Then came along John Bell's little paper. It is a paragon of elegance. Trust me in saying that even if you only had a month of undergraduate quantum mechanics, you could understand the whole thing perfectly -- it is that simple and that well written.
Bell asks a beautifully stupid question. If there is some hidden variable or set of variables -- call them λ -- then we can calculate the probability distribution of those variables, call it p(λ). What's the distribution of observed spins given p(λ)? That's easy to calculate, so he does so. He then shows that the resulting probability distribution for measurements of a pair of entangled particles will be different from those quantum mechanics would predict, regardless of what p(λ) looks like -- a stunning and unexpected result!
People then went off into their labs, measured such systems, and discovered that, lo and behold, it appears the results follow quantum mechanics' predictions, not those that we would get if there's some classical process with a hidden variable. That means there are no hidden variables (at least not local ones, I'm sure someone will pipe up on that!) out there governing how the world works.
For better or worse, quantum mechanics is not an "incomplete theory", and the world around us really does have some sort of irreducible randomness in our observations.