Einstein and his students Podolsky and Rosen were unhappy with quantum mechanics. (QM) 'It's not deterministic' they said. 'It's indeterminate' they complained. (I'll come back to this difference later) They developed a number of thought experiments that they claimed 'refuted' QM. Many of these were shot down by Nils Bohr, Einstein's arch nemesis in these debates. One that stuck, however was the one that became known as EPR, after its creators.
The original EPR involves the uncertainty between position and momentum, which is very tricky. An easier example due to Bohm which captures the problem but is put in terms of spin. This makes it much easier to describe.
Some background knowledge; some particles have spin. No one actually knows what 'spin' means in this sense so let's ignore that aspect. What is important here is that we can measure spin and we get the answer '1' or the answer '-1', we call these 'up' and 'down' respectively. We have to measure spin in a direction. This is to do with the orientation of the Stern-Gerlach apparatus in the experiment, but that is irrelevant. Basically we have a black box that measures spin. And this box has a switch on it. It can either measure spin in the x direction or in the y direction. There is an uncertainty relation between these two spins much like that between position and momentum.
Imagine we have a combined system of two particles that has the property that the sum of the spins in any direction is 0. This is actually possible to do in experiment. So what this means is that if one of the particles is up in the x direction (Ux) then the other is down in x (Dx) and if one is Uy the other must be Dy. Now let's fire our particles in different directions, each at a spin measuring device (a black box). If we switch one box to measure in the y direction and measure Uy then we can know that we got Dy at the other end. If, on the other hand, we had switched our box to x and got Ux then we would know that the other box registered Dx. So there is a fact of the matter about a particle's spin in both directions. This contradicts the uncertainty principle.
And that is the EPR paradox. It shows that QM is not able to measure all the quantities in the world that exist. The theory is incomplete.