Alright let me give this a try...
Quantum entanglement is when the quantum state of two or more objects (photons in this case) are linked in such a way that observations on the quantum state of one affects the quantum state of the other, regardless of physical seperation and that the changes occur instantly regardless of distance (no regard for the speed of light).
When a photon passes through matter (like a crystal for example), it will be absorbed by an electron. Eventually the electron will spontaneously return to its ground state by emitting the photon. Sometimes the photon will instantly decay into two photons with a lower wavelength (and lower energy, energy can't be created). This photon pair is considered entangled.
Because of quantum mechanics, one photon will be horizontally polarized and the other vertically, though we have no way of knowing which one is which. And actually until observed both photons have both polarizations.
If we place a polarization filter in front of one of the photons, it may or may not pass through. If it does, then the other photon will not because the instant a quantum state (polarization in this case) of the first photon is known the second photon's quantum state is created/decided. So even though we haven't made a direct observation on the second photon we have caused it to collapse out of its state of being in both polarizations at once by simply observing the state of its entangled twin.
So the idea was if I could set up a device so I could take two entangled objects, seperate them, then change the quantum state of one I can detect the quantum state of the other and somehow work out a way to transfer information instantly.
Of course this isn't possible. Relativity states that information cannot be transferred faster than the speed of light, otherwise causality could be violated. Fortunately (so we don't unravel the universe I guess) because of quantum mechanics quantum entanglement is useless for transferring information by itself.
But what they can do is transfer the quantum state (teleport) over the distance with the aid of a classical communication channel (that's the fibre optic cable).
Take our two photons with the unknown polarization again. Alice has one photon, and Bob the other photon. Alice can't know the polarization of her photon, and if she observes it that will force Bob's polarization two become the opposite. But since neither "controls" the polarization no information can be transferred that way.
But, Alice can do some manipulation of her photon with the unknown state and send two bits of information to Bob via the fibre optic cable, and with that Bob can recreate the unknown state of Alice's photon, then measure it. In effect Alice's photon has been teleported to Bob (Alice's unknown state is destroyed in the process, since there is a "no cloning theorem" that forbids the creation of identical copies of an arbitrary unknown quantum state).
Doesn't sound too fancy, but it has big implications in quantum computing and quantum cryptography. In this case it's simply a photon, but a qubit (which can hold a lot of information) could be teleported instantly. In fact they've teleported beryllium ions confined in ion traps.
Quantum computing is exciting because while processors we use now operate in a linear fashion, a quantum computer would operate in parallel. Instead of a bit being a 0 or a 1, a qubit is a 0, a 1, or a 0 AND a 1 at the same time. Since qubits can be in multiple states at once, you could perform logical operations on those multiple states all at the same time.
Think of trying to guess a password. Instead of trying aaaaa, aaaab, aaaac, aaaad... bbbba, bbbbb, bbbbc.. etc etc.. a quantum computer would try aaaaa through zzzzz all at the same time (because the qubits would be storing aaaaa through zzzzz at the same time), and the true password would be detectable instantly.
For lots of good reading (though deep) here's the wiki for it:
http://en.wikipedia.org/wiki/Quantum_computer
