Here's the nutshell version: Two subatomic particles, e.g. electrons, can be produced from the disintegration of another particle. These daughter particles then travel at enormous velocities away from each other, but they somehow remain connected such that if a property of one of them is changed the same property in the other one changes simultaneously even though any signal sent from one to the other would have to travel at infinite speed to affect the second. This, though, is impossible, so how does the second electron know what's happened to the first? No one knows the answer to this which is why Einstein, who could never accept the idea of entanglement, called the phenomenon "spooky."
Here's an excellent 15 minute video featuring physicist Brian Greene explaining this quantum weirdness: An article at Nature discusses a recent test that pretty much clinches the theory that somehow particles that are widely separated from each other, even at opposite ends of the universe, are still in some mysterious way connected so that they can communicate instantaneously with each other:
It’s a bad day both for Albert Einstein and for hackers. The most rigorous test of quantum theory ever carried out has confirmed that the ‘spooky action at a distance’ that the German physicist famously hated — in which manipulating one object instantaneously seems to affect another, far away one — is an inherent part of the quantum world.The recent experiments cited in the Nature article are said to show that Einstein was wrong and that entanglement exists. The universe is indeed a very strange place.
The experiment, performed in the Netherlands, could be the final nail in the coffin for models of the atomic world that are more intuitive than standard quantum mechanics, say some physicists. It could also enable quantum engineers to develop a new suite of ultrasecure cryptographic devices.
“From a fundamental point of view, this is truly history-making,” says Nicolas Gisin, a quantum physicist at the University of Geneva in Switzerland. In quantum mechanics, objects can be in multiple states simultaneously: for example, an atom can be in two places, or spin in opposite directions, at once. Measuring an object forces it to snap into a well-defined state. Furthermore, the properties of different objects can become ‘entangled’, meaning that their states are linked: when a property of one such object is measured, the properties of all its entangled twins become set, too.
This idea galled Einstein because it seemed that this ghostly influence would be transmitted instantaneously between even vastly separated but entangled particles — implying that it could contravene the universal rule that nothing can travel faster than the speed of light. He proposed that quantum particles do have set properties before they are measured, called hidden variables. And even though those variables cannot be accessed, he suggested that they pre-program entangled particles to behave in correlated ways.