In 1935, Einstein, Podolsky and Rosen published a paper in which they imagined a photon which emits an electron and positron which zoom off in opposite directions as two entangled particles. Quantum mechanics says they do not have definite spin. If you then measure the spin of one of them when they are far apart, and it happens to be spin up, then the other particle must be spin down. Since the two particles suddenly have these definite spins, information must have travelled instantaneously between them over a potentially huge distance: violating special relativity. This unnerved Einstein because he, by then, supported the idea of "local realism". Local means that information cannot travel faster than light and realism says that the Moon is there even when you can't see it. The young Einstein did not agree with realism. His special relativity came from saying "if you can't observe it, it doesn't exist", quantum mechanics relies on this too, but the later Einstein and others argued that the required spin information is always there hidden inside the entangled particles (as a hidden, dark, variable) and is only revealed upon measurement.
However, in a 1964 paper John Bell identified a measurable experimental difference that would occur depending on whether local realism (the hidden variable) was true or not. If you measure the spin of the two particles at random angles, then if you happen to measure them parallel you will get anticorrelated results, and you measure perpendicularly you will get no correlation, but at intermediate angles, if local realism was true there would be a linear dependence of correlation on the angle, if not there would be a cosine dependence. In the 1970s and 1980s Freedman and Clauser and Alain Aspect and others measured this, and found a cosine dependence. So it looks like the common sense thing: local realism, does not work. It is true that all of the experiments that have been done so far have had loopholes in them that might allow them to still be explained by local realism, but they have all had different loopholes, and so you would need a different version of local realism to explain each case, which seems unlikely. A loophole-less experiment would be conclusive, but may not be possible.
As Heisenberg always said "How fortunate that we have found a paradox. Now we have some chance of making progress!". These experiments support a philosophy that has always seemed to work, and was used by Berkeley, Mach and the younger Einstein, that if things can't be seen "in principle" then you have to assume they do not exist. I also like this philosophy. In my recent paper (EPL, 101, 59001, arxiv preprint: 1302.2775) I argued that the inertial mass of an object is caused by Unruh radiation and that, for example, when an object accelerates to the right, a Rindler horizon forms to its left because objects behind the horizon can never hope to catch up so their information is lost to the object. This means (to simplify) that Unruh waves longer than the distance to this new Rindler horizon can never be seen by the object and therefore, with this philosophy, don't exist as far as it is concerned. As a result of this, there is less Unruh radiation to the left of the object, less radiation pressure on it from the left and this produces a net force towards the left that opposes the acceleration. This works neatly as an model for inertia (see 1302.2775). All this relies on the principle that if something cannot be seen in principle, it dissapears, or: undergoes total existence failure (a amusingly over-complicated term used by Douglas Adams).
Einstein once said to Abraham Pais (who did not believe in realism): "Do you really think that the Moon doesn't exist when you are not looking at it?". My answer would be that it is there if you don't look, but if it is impossible for you to look (a fundamental horizon gets between you) then it would not be there, or more importantly: its effects would not be. I think that this can be tested, more on this later..
Einstein, A.; Podolsky, B.; and Rosen, N., 1935. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete. Phys. Rev., 47, 777-780.
Bell, J.S., 1964. Physics 1, 195-200.
McCulloch, M.E., 2013. Inertia from an asymmetric Casimir effect. EPL, 101, 59001. http://arxiv.org/abs/1302.2775