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