OK - I'll see what I can do with an outline on entanglement. Then I'm getting some sleep.
Entanglement is a counterintuitive result which arises from a seeming collision between several laws of physics that are well-known even to high school physics students: (1) Conservation of [energy, momentum, charge, angular momentum, etc.], (2) That information cannot travel faster than the speed of light, and (3) the superposition of states and observer effect that we just discussed
We know that just as an electron and positron (anti-electron) can annhilate and produce a pair of photons, similarly a pair of photons can produce an electron-positron pair. Electrons and positrons have spin (angular momentum with some restrictions; let's call it "up" or "down"). By conservation laws, we know that the total angular momentum will remain 0 after the pair of particles is produced. So, if we produce an electron-positron pair, measure the spin of the electron, and then measure the spin of the position, we should expect one to be up and one to be down. But, we don't know which is which before measuring. So, let's consider three experiments. [Note: Yes, I'm slightly lying in the way I've presented a couple of these because the wavefunction can spread again; feel free to PM me if you want to quibble and I'll give you a more rigorous answer if you're willing to do some math]
Experiment 1: Produce an electron-positron pair, and move them a large distance apart. Emily measures the spin of the electron, and tells Paul her result. Paul then performs a test that measures whether the position is spin-up, spin-down, or in a superposition of both states.
Result: On average, Emily gets 50% up and 50% down, as expected. In our example, let's say Emily turns out to get an electron that is spin-down. Paul now performs his test, and discovers that the positron is spin-up. This is what we expect, because of conservation of angular momentum.
Experiment 2: Emily does not measure anything about the electron. Paul then performs a test that measures whether the position is spin-up, spin-down, or in a superposition of both states.
Result: Since nothing has been measured, Paul finds that the positron is in a superposition of both states. This is the result we expect from quantum mechanics.
Experiment 3: Now it gets tricky. Emily and Paul synchronize their clocks. Emily measures the spin of the electron. In this case, it's spin-up. Paul is a long distance away, and before information travelling at the speed of light would be able to travel between Emily and Paul, Paul performs a test that measures the spin of the positron. From a relativity point of view, this means that neither Emily's test nor Paul's test happens *before* the other, and neither one should be able to affect the other.
Now, we have a problem. What results do we expect? Well, conservation laws tell us that Emily and Paul should get opposite results. So, one will always be spin-up and the other will always be spin-down.
However, quantum mechanics tells us that before measurements are done, the electron and positron are in a superposition of both states. And, relativity tells us that the measurements done by Emily and Paul cannot affect each other. Thus, Emily and Paul should each get a random result, which means that 50% of the time they should have opposite spin and 50% of the time they should have the same spin.
Actual result: Emily and Paul always measure opposite spins. In other words, the conservation laws are correct, but either relativity or the superposition of states is violated. Since we tested the superposition of states (Experiment 2), it appears that relativity is wrong -- somehow, information is indeed travelling faster than the speed of light. Experimentally, a minimum of something like 10^4 times faster than light, and possibly instantaneously.
Note that this isn't unverifiable; in fact, the reason we've adopted this strange theory of quantum entanglement is precisely because as strange as these results are, they're experimentally verified.