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.
Additional notes: (1) The given states of the two particles are prepared at O. But no one knows what they are until they are measured at A or B, and it doesn't matter which one is carried first or second. The states were already determined at O. Now, when we write our wavefunction, we do it as a superposition as we don't know what the states are, that is, prior to the measurement. Our writing the wavefunction as such is a mark of our ignorance. Just like tossing a coin, prior to its observation of what the outcome will be, we can only say it's either heads or tails, and we know it's 50% heads, 50% tails. But once the outcome is observed ( the coin has landed in our hand), we know what it is ( heads or tails). Similarly,before the spins are measured, we can write:
ψ = 2-1/2( | ↑↓ > - | ↓↑ > )
Notice we write this purposefully because we know that there is a 50% chance the spins will be up at A, down at B (first term in the bracket), and 50% chance down at A, up at B (second term in the bracket). Those who claim that the particles EXIST in those superposition states before the measurements are fabulating as they don't have that knowledge - the coin doesn't exist in a superposition while it is being tossed in the air, it will yield its outcome of heads or tails position when it is forced to come to rest. Similarly, the particles will yield the up/down spin when they are forced to pass through a magnetic field. Hence the above equation is not about existence but that it contains information (in this case, the possibilities that can occur at either A or B) and thus allows us to calculate probabilities. To talk about "wave collapse" and "instantaneous transfer of information from A to B" (or vice-versa) is totally off the mark.
(2) Entanglement is useful since once we know the spin of one particle, we automatically know the spin of the other particle - in the example above, this is entirely due to the conservation of angular momentum.