Quantum mechanics cannot tell you precisely where a particle is located, only where it probably is. Does that seem a little strange to you? It also says that two particles that share a history can interact with each other no matter how far apart they are. These 'entangled' particles appear to interact with each other instantaneously, without any transmission of information, despite the speed of light being the fastest possible speed. When one particle is measured, the other one automatically adjusts itself to correspond to the first one's measurement. Does that mean that something is traveling faster than light? Einstein was adament that something was missing, and that quantum mechanics as a theory is incomplete.
Einstein called this 'instantaneous' interaction of separate particles "spooky action at a distance". It turns out that what troubled him is not a debatable point, but an actual observed phenomena. This actually happens!
Suppose I have two marbles: one blue and one red. I put each of them in two individual opaque packets, and send one to Japan and the other to Egypt. If you open the packet in Japan and find that the marble inside is blue, then you know right away that the one in Egypt is red. The colors of the marbles were already predetermined, and the color information was already present even though it was hidden from us behind the opacity of the packets. Sounds about right, doesn't it? Einstein argued that this must be the case with entangled particles: there must be some hidden variable that can explain the whole thing. Since quantum mechanics doesn't allow these local hidden variables, he argued that it was incomplete.
Let's take a look at a different scenario, one allowed by quantum mechanics: consider two magical, colourless, identical marbles, that become coloured if someone touches them. They are 'entangled' in such a way that if one turns red, the other definitely turns blue. Each individual marble is as likely to turn red as it is to turn blue, at a single touch, but once you touch it, the other one has to take on the opposite colour. How does the second marble know what colour the first marble is? There was no predetermination. There are no hidden variables. The second marble doesn't say "okay you go red, and I'll go blue", and the first one doesn't send the second one a message that is faster than light. Before the first marble was touched, it was happily lying in a state of ignorance about whether it would turn blue or red. It knew that it had an equal probability of turning either colour. But, somehow, because the two were magically 'entangled', the colour of one influenced the colour of the other.
Seems a little un-intuitive, doesn't it? But this actually happens! This has actually been observed in the lab with photons and electrons and other systems of entangled particles. And physicists are still working out the implications.
Does this make you uncomfortable? It certainly discomforted Einstein.
More to read:
Wikipedia -- Quantum Nonlocality, EPR Paradox, Bell's Theorem
Is the moon there when nobody looks?
Bell's Theorem With Easy Math
Does Bell's Inequality rule out local theories of quantum mechanics?
More complicated: A Historical and Modern View on Bell’s Inequality