Quantum entanglement is when two (or more) quantum particles or waves relate to each other, as in taking spins opposite to each other. When they are separated they retain the same opposite spins, and the observation of the spin of one of the states predicts the opposite spin for the other state, even though the quantum states or particles may have been separated by many miles from each other.
The implications of this phenomenon are enormous, and form the basis for the new field of quantum computing. For example, there seems to be an instantaneous communication between the participles at the moment of observation of one of them. Some physicists resist that notion and claim that no information is actually communicated.