Quantum entanglement is when two (or more) quantum particles 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 determines the opposite spin for the other state, even though the 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 particles at the moment of observation of one of them.
On the other hand, some physicists resist that notion and claim that no information is actually communicated. This is compatible with the theory of relativity, since it is not possible for someone to transmit information faster than the speed of light by somehow encoding it with entangled particles.
Quantum mechanics says that there is a fundamental uncertainty in nature due to the Heisenberg uncertainty principle. It is not just that we are only able to measure a particle's position, for example, to a limited accuracy, but the particle itself does not have a precise position. Local hidden variable theories propose that particles do in fact have precise position, but observers have a limited precision available to them.
At first sight, it appears that these two theories are indistinguishable from each other, but this is not true. In 1964, John Bell published his famous Bell's theorem, which showed that it local hidden variable theories and quantum mechanics do make different predictions. Hence it is in fact possible to perform am experiment to distinguish which theory is correct. Experiments have shown that quantum mechanics is in fact correct, and that the entangled particles do "communicate" faster than speed of light. However, non-local hidden variable theories are still possible. These theories do not have the requirement of locality, which prevents faster than light communication.