The EPR-paradox, as it is commonly known as, is a thought experiment targeted at the philosophy of quantum mechanics, at that time represented by the Copenhagen Interpretation

It was published as an article in 1935 written by Albert Einstein,  Boris Podolsky and Nathan Rosen. 

The thought experiment showed how two electrons, first joined closely and then separated by a great distance, can affect each other instantaneously. More in detail, it had to do with how a two-electron system with spin angular momentum. Measuring this on one electron, you immediately know what the other electron has, since the sum has to be conserved. The paradox is that when you measure the angular momentum along one of the (x,y,z) axis, you also define the state of the electron. What state depends on what axis you measure relative to. Therefore, one electron can get a defined state, when another electron light-years away has its angular momentum measured. The conclusion of E-P-R was that there must be something incomplete about the quantum mechanics and that there had to exist what they called "hidden variables". "God doesn't play dice" was what Einstein said about this, meaning that there had to be something that we couldn't measure.

This thought experiment was further examined and refined by John Bell in 1964, leading to Bell's Inequality. In short, he showed that if there were any hidden variables, then the correlation for the values of measurements as mentioned above had to follow a certain quantitative relationship.

Experiments performed later broke Bell's Inequality and complied with quantum mechanics, proving that there are no hidden variables.

In 1982 French scientist Alain Aspect and his team formed a experiment that actually simulated the one suggested by E-P-R. They actually measured two particles in such a short time interval that no information could be transferred between them - according to Einstein's Special Theory of Relativity. The result was that the measurement of one particle clearly defined the state of the other. Since it was already proven that there are no hidden variables, and no information was exchanged, the conclusion was that the original Copenhagen Interpretation is correct. The wave/particle duality is real in the physical world.

Source: ne.se, my own head after reading too many physics books, britannica.com for English terminology

The result of a thought experiment invented by Albert Einstein, Boris Podolsky and Nathan Rosen and publised under the title Can quantum-mechanical description of physical reality be considered complete? in issue 41 of Physical Review in 1935. It was intended to show contradictions, or at least a lack of completeness in quantum theory, which could only be resolved by adding additional hidden variables. The experiment works like this:

1. Take two physical systems that initially interact with each other in such a way that both are described by a single Schrödinger wave equation. This allows you to infer information about one system from an observation made on the other.
2. Separate the systems
3. Make a precise measurement in one system that allows you to infer information about one in a set of noncommuting observables in the second system.
4. Make a precise measurement of one of the other observables of the set in the second system.
5. Et voila! You have just violated the uncertainty principle in that system.
6. Watch the universe evaporate in a cloud of logic.

To be more specific: in step 1, take two photons that emerged from the decay of a neutral pion at rest. The law of conservation of momentum says that they must have exactly opposite momentum, since the original particle had none. In step 3, measure the momentum of the first photon. You now know the momentum of the second one without having actually touched (measured) it. In step 4, measure the position of the second photon, which the uncertainty principle says you cannot know at the same time as the momentum.

It counts as a thought experiment because current technology is unable to create completely isolated physical systems, let alone separate and move apart two that contain particles moving at the speed of light. Basically, any straightforward setup would fuck up the measurements many times over.

However, in 1964 John Bell formulated a derived formula (now called Bell's Inequality Principle) which would show the existence of the above-mentioned hidden variables and could be tested for in a variety of far more manageable circumstances. Results were negative, but many physicists argued that there are subtle flaws in Bell's reasoning or that the experimental setups were not adequate. The issue remains unresolved.

The Einstein-Podolsky-Rosen paradox is that measurement in quantum mechanics seems to require faster-than-light communication under certain circumstances. This, they claimed, is absurd, and proves quantum mechanics to be incomplete -- certain things must be decided in advance.

This argued towards a local hidden variables theory: a theory which maintains locality better than quantum mechanics and had the 'side benefit' of not being so weird (though this was what Einstein, at least, was really after). How was it less weird? The distributions of the properties of particles were all now statistical distributions over unknown quantities, and not intrinsic distributions over quantum operator eigenvalues.

At first, there was no known way to actually test whether the seemingly absurd result of faster-than-light communication was true or not: no one could think of a prediction quantum mechanics made that no local hidden variables theory could make. In 1951, some progress was made as David Bohm created a more tractable variant of the paradox involving spin, but still no specific different predictions could be found.

In 1964, this changed: John Bell used Bohm's special case to devise his famous inequality, which pointed out a difference between the two schema. That is, though there were a variety of ways a local hidden variables theory could make things work, there was a limit to how coordinated they could make things. Quantum mechanics could cross this limit.

Alain Aspect used a special case of the inequality to form the basis of his famous experiment, which was finished in 1982.

The final form of the Aspect experiment went so:

1. Set up a device which creates Einstein-Podolsky-Rosen pairs (EPR pairs) of photons proceeding in opposite directions. What makes each pair an EPR pair is that the two photons' polarizations are oriented the opposite way*, not merely coincidentally, but they are forced to be so. This quantum dependence is known as entanglement. Further, we do not know which way the photons' polarizations are oriented.
   To get technical, the spinor ket of the photon pair is A | + - > + B | - + > for some A and B, in a linear polarization basis.
2. Place three detectors to detect each photon. Have each detector only detect along one of three axes. Distribute these axes equally around the circle. That is, give them a sixty degree angle of separation.
3. Rapidly randomly determine which axis is used on each detector, and reset the choice after each photon is detected. Make sure the switching is good enough to keep the switching events spacelike separated.

There are two cases here: the photons' polarization is detected along the same axis, or the photons' polarization is detected along different axes. In the event that the photons were detected along the same axis, things are simple -- they will be read oppositely. This serves as a check on the efficiency of the setup.

In the other case, in which the photons are measured along different directions, the Bell Inequality comes into play: quantum mechanics and local-hidden-variables theories make different predictions on how often the two spins will look 'more opposite' than 'more aligned' (recall that the detectors are not aligned, but they are not orthogonal either).

As it turned out, quantum mechanics' prediction was strongly validated.

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Even after this experiment, there were a few loopholes through which it was conceivable that one could fit a local hidden variables theory: the theory could involve 'looking ahead' at the detectors and finding what orientations they would be, going so far as to examine the state of the randomization mechanism. With improving randomization, this became an increasingly wild supposition. As time progressed, the various loopholes were closed tighter and tighter: supposing local hidden variables now requires incredibly complicated and un-physics-like 'conspiracy'-style theories.

Taking this to mean that local hidden variables theories are false, what does that leave?

• Quantum mechanics
  Up side: we already know what it is, it has succeeded all tests.
  Down side: under the Copenhagen interpretation, locality is violated.
• A global hidden variables theory
  Up side: at least Quantum mechanics and all of its weirdness isn't true.
  Down side: we don't know what such a theory would be (though one has been devised, by David Bohm), and since there isn't a single difference in testable predictions between any of them and quantum mechanics, pursuing it has entered the realm of metaphysics. To snag a quote from a Nobel-Winner** "If it makes different predictions from Quantum Mechanics, I'm not interested. If it makes the same predictions as Quantum Mechanics, I'm not interested."

At first, it seems like we're stuck with nonlocality in our physical theory, whether by global variables or by a global wavefunction which collapses superluminally. This would be downright ugly. But the locality problem is not with Quantum Mechanics itself, but with the Copenhagen Interpretation: it is the collapse of the wavefunction which is a problem. If we consider the measurement process to be another case of entanglement, then the consistency of the results follows straightforwardly and involves only local information exchange -- the exchange occurring when you bring together the various results (note that this one supposition is the entire basis of the Many-Worlds Interpretation).

Entirely separate from the philosophical implications, this paradox yielded a tool of practical utility: EPR pairs form the basis of Quantum Teleportation, and play a major role in Quantum Computing.

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* An entangled system can have any sort of relation, not only opposite spin. The relationship does not even have to deal with spin. There can be more than two particles, and they do not even need to be the same type. The case used in the experiment kept things as simple as possible.

** I believe it was John Schrieffer, but I could be wrong; it's hard to track these things.

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The papers:

Einstein, A.; Podolsky, B.; and Rosen, N. "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Phys. Rev. 47, 777-780, 1935.

Bohm, D. "The Paradox of Einstein, Rosen, and Podolsky." Quantum Th., 611-623, 1951

Bell, J. S. "On the Einstein-Podolsky-Rosen Paradox." Physics 1, 195-200, 1964.

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.