Quantum entanglement is the phenomenon of two (or more) particles relating to each other in a certain manner under the special rules of Quantum mechanics, 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 particles always obtains the opposite spin for the other particle, even though they may have been separated from each other by many miles. This phenomenon lies outside of normal intuition. Physicists refer to the normal state of affairs as "local realism". Strict quantum mechanics is sometimes at odds with local realism, and entanglement is an example of that.
Experiments show that when local realism conflicts with strict (Copenhagen) quantum mechanics, the latter is correct.
The implications of this phenomenon are enormous, and form the basis for the new field of quantum computing.
When the two particles reveal their states at their different locations, and those states always correspond, it might seem that there must have been some kind of instantaneous communication between them, and that this would constitute supraluminal information transfer. However, this "transfer" lies outside of local realism. There is no actual transfer that anyone could observe or exploit.
Some physicists 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 an experiment to distinguish which theory is correct. Experiments have shown that quantum mechanics is in fact correct, and that the entangled particles do exhibit their "entangled" information at locations sufficiently far apart that, if information were actually transferred between them, that transfer would have to be faster than the speed of light. However, no actual "communication" occurs between the particles. But non-local hidden variable theories are still possible. These theories do not have the requirement of locality, which prevents faster than light communication.