Entanglement, Bohr-Einstein Debate, Bell's Inequality On BBC episode The Secrets of Quantum Physics (Part 1) Jim Al-Khalili explains quantum mechanics for the layman. In the first half, he does a very good job; in the second half, either he thought his explanations/analogies were clear but were not, or I terribly missed something. Here is a summary of the part on “entanglement”.

Let me try and explain this by imagining the two particles are
  spinning coins. Imagine these coins are two electrons created from the
  same event and then moved apart from each other. Quantum mechanics
  says that, because they're created together, they're entangled. And
  now many of their properties are for ever linked, wherever they are.
  The Copenhagen interpretation says that until you measure one of the
  coins, neither of them is heads or tails. In fact, heads and tails
  don't even exist. And here's where entanglement makes this weird
  situation even weirder. When we stop the first coin and it becomes
  heads; because the coins are linked through entanglement, the second
  coin will simultaneously become tails. And here's the crucial thing: I
  can't predict what the outcome of my measurement will be, only that
  they will always be opposite. Einstein believed there was a simpler
  interpretation: Quantum particles were nothing like spinning coins;
  they were more like, say, a pair of gloves, left and right, separated
  into boxes. We don't know which box contains which glove until we open
  one, but when we do, and find, say, a right-handed glove, immediately,
  we know that the other box contains the left-handed glove.

[To compare these two theories, Jim Al-Khalili devises a semi-analogous card game.]

The card game is against a mysterious quantum dealer. The cards he
  deals represent any subatomic particles, or even quanta of light,
  photons. And the game we'll play will ultimately tell us whether
  Einstein or Bohr was right. Now, the rules of the game are deceptively
  simple. The dealer's going to deal two cards face down: If they're the
  same color, I win; if they're different colors, I lose.

[He loses each game.]

I know what the dealer's doing here. ,Clearly, the deck has been
  rigged in advance so that every pair came out as opposite colors. But
  there's a simple way to catch the dealer out. So what we can do now is
  change the rules of the game. This time, if they are the opposite
  color, I win.

[He loses each game.]

I'm now not going to tell the dealer which game I want to play, same
  colors wins, or different color wins, until after he's dealt the
  cards. Now, because he can never predict which rules I'm going to play
  by, he can never stack the deck correctly. Now he can't win...or can
  he? This gets to the very heart of Bell's idea. If we now start
  playing and I win as many as I lose, then Einstein was right. The
  dealer is just a trickster with a gift for slight of hand.

[He loses each game.]
Assuming that the analogy is correct (who am I to judge him), how does the third card game relate to the Einstein-Bohr debate? Specifically, his conclusion, “This gets to the very heart of Bell's idea. If we now start playing and I win as many as I lose, then Einstein was right”.
Continuing, Jim Al-Khalili performs an experiment, which supposedly proves that Bell's Equation was right. Is there a nontechnical explanation of how the truthfulness of Bell's inequality confirms/proves Bohr's interpretation on “entanglement”?
 A: Bells inequality doesn't prove that Bell or Einstein are right, it only shows (mathematically) that there is a difference (an inequality) in the predicted outcomes of Quantum Mechanics and Classical Mechanics even before an experiment is done. Classical mechanics relies on local hidden variables and Quantum mechanics more on Spooky action at a distance so to say. Bell's theorem on the other hand states that no real or physical local hidden variable theory can reproduce all the predictions of quantum mechanics. The third card game is an analogy to represent some of the actual experiments where two entangled particle (which are very hard to produce) are tested for correlation. Two entangled photons fired are at two separate detectors but along the way they go through a slit that can be set in three different positions 120 degrees apart. The photons will (Y) or will not (N) go through the slits. Bell's theory was that no matter how you set the slit settings there will only be eight different outcomes. Depending on the photon and the settings the photon may not go through (theoretically) any of the slits NNN or all of the slits YYY. Then there are six other combinations of NNY, NYN, NYY, YNY, YYN, or YNN. The other entangled photon at the other detector has the same odds and when you compare the outcomes Bell's math claims that the matching corelation will be no more than 33%. On the other hand quantum mechanics experiments test 25%. Sorry I may be off on the way I'm remembering the percentages but I think that's the basics.    
A: I just watched the video and here is my understanding.
First the card game is just an analogy, which is not literally a physical phenomenon. Once this being understood, we can go ahead to see how each part of the game corresponds to that in the Bohr-Einstein debate, the so-called EPR "entanglement game".
In the EPR entanglement, there are a pair of particles, correlated to each other by the initial interaction. Because of the random nature of the system, each particle could be found in either of the two states: up or down. But the states of each pair of particles are fully correlated. The moment when the particle 1 is found to be up, the particle 2 is down. Vice versa, the particle 1 down and the particle 2 up.
There must be some underlying model of the unseen nature to explain what was observed, namely, the correlation between the two particles in each pair. The card game corresponds to the experiment for determining which model might explain what was observed. The first two card games corresponds to the two possible combinations described above, namely,


*

*particle 1 up and particle 2 down

*particle 2 down and particle 1 up
Here the Jim/two-card correspond to particle 1/2. The up/down corresponds to the same/different color. The dealer corresponds to the unseen nature. In these two games, what was observed could be easily explained by the initial rigging action of the dealer. This explanation corresponds to the Einstein's model, in which what was observed had been arranged in advance by the dealer/nature. In the Bohr's model, it states that the dealer hadn't rigged the deck in advance, the correlation always happened at the moment when Jim decided what to choose (the same or different color). Therefore the third game came into the play to check if the dealer had really fixed the cards in advance. If it had indeed done so, which would have proved Einstein was right. Otherwise, it would have proved Bohr was right. Surprisingly, Bohr was right, as shown in the video. It appeared that the dealer didn't fixed cards in advance. The correlation could only be explained as if it occurred the moment Jim decided what to choose.
The heart of the whole game analogy says that, the quantum correlation is much stronger than the classical correlation.
A: Let's say the particles in your first paragraph are pairs of photons, one of the choices the TV program considered. Then head means a photon is polarised in some given direction and tail in another given direction. We need two devices called analysers to measure each polarisation. One probes one photon and the other one the other photon. So now, in this card game analogy, the colour of the cards is a mashup of both the polarisation of the photons and the setting of those analysers, I think. So this makes it hard to precisely relate the games to real experiment. But I can still get the gist of the last game I think. It represent experiments where the settings of the analysers is changed while the photons are in flight.
This is done to defeat the possibility that card deck was rigged beforehand, since now the card dealer cannot predict the player choices, exactly as the mechanism producing the pairs of photons cannot know, or however the pair of photons change during their flight, it/they cannot know what the settings of the analysers will be when they get to them.
To really answer your last question, the game considered in the program is too confused, as I pointed out above. The example of the boxes with gloves they started with can be spun in a more satisfying way actually, and I don't understand why they did not pursue this course. I can do that later if you wish.
A: The bottom line is that it is really tough to explain what quantum entanglement is and is not. But I can tell you the gist of what Bell proved. Bell proved that a certain model of the physical world, a very general model and, one would think, natural one, does not agree with quantum physics. So experimenters did the check and it turned out that the type of model discussed by Bell (called local realism) is ruled out, whereas quantum physics is not ruled out.
The model called local realism basically says that each part of the physical world can be considered to be described by information that is located wherever that part is (information such as its mass, charge, density, spin direction and so on), and which does not depend on what is going on in places not connected to the one under discussion (to make this specific you can say spacelike-separated places). This is the model assumed pretty much throughout science---except in quantum physics where it turns out to be not quite right.
The tricky bit is presenting the evidence for this conclusion. That is what Jim Al-Khalili  was trying to do with his card game examples. The cards represent the idea of information contained or written down on the physical object (the card). Dealing them represents placing objects at different places where they no longer interact with one another. Looking at them represents experimental observation. And what happens, if you choose your experiment carefully, is that the results are not consistent with the notion that the information is printed on the cards individually. It is more like a case where a pair of cards has the property "we are of the same suit" but without either of them having a suit of its own. When a third party interacts with one card of such a pair, it cannot help but interact with them as a pair, even though they may be separated. But this form of interaction does not amount to action-at-a-distance, because what happens is that the third party shares in the correlation, and correlation can only have observable effects when the two (or more) correlated parts are brought together.
A: It's funny how people keep saying "the Copenhagen Interpretation says such-an-such and that's really weird and makes no sense". Such an embarrassing string of bad explanations really ought to make one think twice about accepting it as an explanation.
There is NOTHING in the equations which says the wavefunction collapses, that's a superfluous idea added by confused physicists in the 1920s and we're still chucking it around. Macroscopic decoherence is the simple, straightforward coherent explanation.
In this picture, the dealer with the cards analogy breaks down completely -- it can't translate and still remain intact.
Going back to a real phyiscal example. Suppose you have two particles created in an entangled state. One is "spin up" and one is "spin down" but you don't know which is which. Here's what happens, the initial situation is that the following statements are both simultaneously true:
1) A is spin-up and B is spin-down.
2) B is spin-up and A is spin-down.
This is your superposition of states. Then you perform an observation of the spin of particle A. The following statements are then both simultaneously true.
1) A is spin-up and B is spin-down and you observe "A is spin-up."
2) B is spin-up and A is spin-down and you observe "A is spin-down."
The states are still superposed, see?
Later you observe the spin of particle B. The following statements are then both simultaneously true.
1) A is spin-up and B is spin-down and you observe "A is spin-up" and you observe "B is spin down".
2) B is spin-up and A is spin-down and you observe "A is spin-down" and you observe "B is spin-up."
The wavefunction of the whole universe contains both outcomes in their entirety, at least that's what the equations say. When you complain "but I only observe A is spin-up, not A is spin-down" what you are observing is which part of the wavefunction "you" are in.
The only problem is understanding what the word "you" means in this description.
