You have two particles that are released at O, sent in opposite directions with opposite spins, moving toward A and B, where observers are stationed. These particles are by definition entangled.
This is the claim: when A makes a measurement of the particle spin, say, it is an up spin. Then he can conclude that B will measure the spin of his particle to be down. Einstein had argued that this would contradict relativity since A knows the spin at B instantaneously, and no signal can travel faster than the speed of light.
The paradox is resolved when you look at the setup prior to the measurement of the spins. The observer at A had to know that another particle was sent to B. How does he know? Well, for one, the person at O releasing the two particles in opposite direction could communicate with A and tell him that a particle was coming his way, and that another was moving in the opposite direction with an opposite spin towards B. If A doesn't have that knowledge, he can't conclude what happens at B since he is not in the position to know that a second particle was released in the first place. So nowhere in this experiment is a signal moving faster than the speed of light. Observer at A happens to know the spin at B because he was told ahead of what O did in preparing the entangled particles. Einstein looked at this experiment from a God's point of view, knowing exactly what happens at O, A and B. And then he erroneously concluded that either a signal would have to travel at speed faster than light or there was a spooky action at a distance. But for an observer actually doing the measurement, you're either at A, B or O, not simultaneously at all three positions.
What the experiment shows is if two particles are prepared in a given quantum state, unless there is an interaction, they will continue to stay in that quantum state. Conservation of energy, momentum, and spin (angular momentum) requires that much. So sending these two particles to the ends of the galaxy, and then say that if one measures the spin of one, he knows instaneously the other spin becomes a mystery only if you adopt God's point of view!
The real mystery is why the internet is flooded with so many sites that contain the most obvious misconceptions about quantum entanglement. There is no spooky action at a distance. There is no need of hidden parameters. And no signal was sent faster than the speed of light. It simply means that if two particles are prepared in a given quantum state, unless there is an interaction, they will continue to stay in that quantum state.
It's time to admit that some of the great scientists were wrong on this issue.