Quantum entanglement stems from the EPR paradox. The textbook example is that you have a reaction that emits two photons in opposite directions. To conserve spin, one has to be "spin up" while the other is "spin down". But you can't know which is which, and both photons are in a superposition of both states. When you measure one, it must "choose" a state, but then the other has to "choose" the opposite state to comply with conservation. If you measure them both at precisely the same moment, they will be in opposite states even though they did not have time to communicate within the limits of the speed of light. Thus they are entangled, and are not restricted by the speed of light to communicate their states with each other.

Originally it was a paradox. But we've advanced to the point that we can actually run that test, and it turns out to be correct.