I think it's worthwhile to make the proof as simple as possible, in part because some people continue to have trouble with it and some continue to produce false "counterexamples" of purported local hidden variable models that violate Bell's inequalities.
The focus here is on the implications of Quantum Mechanics itself. I will not discuss here practicalities of experiments to test those predictions, such as "loopholes" due to low detector efficiencies. Such loopholes are implausible and more recent experiments have closed some or made them even less plausible.
Any entangled pair of systems, each with at least two distinguishable states besides for position, could be used. I'll use an entangled pair of spin-1/2 particles for ease of notation.
A pair of spin-1/2 particles are generated which are in the entangled spin singlet state
|psi> = (|+,-> - |-,+>) / 2^(1/2)
One of these particles is sent to Alice; the other is sent to Bob. The two observers may be very far apart.
Alice <----------------- source ---------------> Bob
When an observer measures the component of a spin-1/2 particle's spin along any direction in space (for example, using a Stern-Gerlach device), the result of the measurement is always + or - 1/2 hbar. (hbar is Plank's constant / 2 pi)
In the state psi> above, there is a 50% chance that the spin component will be positive (+) for any direction of measurement for either particle.
In the state psi>, QM predicts that if Alice's particle is measured with the result + in direction A (call this A+, etc.), then measurement on Bob's particle in the same direction A will give the - result, A-. This kind of correllation is called an EPR correllation.
Call this pair of results (A+; A-) where A is the direction and the order indicates which Observer gets each of the results.
Let P(A+; A-) be the probability that (A+; A-) is found, and so on. Clearly, P(A+; A-) = P(A-; A+) = 1/2. There is no need to consider models that don't predict this for the proof, since they already would disagree with the predictions of QM.
Each of the Observers can choose which direction to measure the spin along. In particular, each will choose one of three directions: A, B, or C.
Define Distant Measurement Independence (DMI) as the assumption that the (singular) result of each Observer's measurement can not depend on which direction the other Observer chose to measure along.
Statement of the Theorem: DMI is not consistent with the predictions of QM.
The theorem is often (incorrectly) said to prove that QM is non-local, because a reasonable local model would not allow the direction chosen for a distant measurement to influence the result of the other measurement. That is not the only local possibility!
If DMI is false there are 3 possibilities, of which the first two are taken seriously:
1) Nonlocality: An instant (faster-than-light) hidden signal which conveys the information about the measurement angle (which can be ‘chosen’ right before measurement) to the other particle, no matter where it is or how far away.
2) Multiple outcomes of each measurement actually do occur (as in the MWI).
If all outcomes occur, correlations might be established only after local interactions; see http://arxiv.org/abs/0902.3827
3) “Conspiracy theories” in which the other particle somehow can predict the angle.
Assume DMI. It is possible that a model assigns certain additional properties to a particular particle that don't appear in the QM description; these are called hidden variables. These could tell the particle whether to give a "+" or a "-" result as a function of what direction its measurement is made in.
The other particle of the pair would then have to have a similar set of properties but with the opposite instructions. Such hidden variables would be required in order to produce the EPR correlations without violating DMI, because otherwise the other particle would have no way to be certain to give the opposite result when both Observers choose the same direction.
Even though only one measurement on the particle is actually made, it can be thought of as labeled by the hidden variables according to what the outcome of measurement along each of the three directions A, B, and C would have been. Consider Alice's particle to be so labeled.
Let (A+ & B-) mean that the result would be + if measured along direction A and would be - if measured along direction B, and so on. Let P(A+ & B-) be the probability that the hidden variables were such that those would be the results.
The following inequality must hold since more general cases are at least as probable as less general ones:
P(A+ & B-) = P(A+ & B- & C-) + P(A+ & B- & C+) ≤ P(A+ & C-) + P(C+ & B-)
It is not possible to measure Alice's particle along more than one direction, but Bob can help us do the next best thing; because of the EPR correlations, measuring his particle should reveal the opposite of what result Alice's particle would have given. Thus the above inequality is equivalent to
P(A+ ; B+) ≤ P(A+ ; C+) + P(C+ ; B+)
This kind of inequality is called a Bell inequality (of which there are actually several).
Quantum mechanically, P(A+ ; B+) = 1/2 sin^2 (theta(A,B)/2) where theta(A,B) is the angle between A and B; and so on.
For example, say A and B are at a right angle, with the C direction in between them.
theta(A,B) = 90 degrees, and theta(A,C) = theta(C,B) = 45 degrees.
Then P(A+ ; B+) = .25, and P(A+ ; C+) = P(C+ ; B+) = .073
Since .25 > .146, the inequality is violated. This establishes that DMI is not consistent with QM.
You can see http://arxiv.org/abs/0902.3827 for another overview of Bell's theorem.