Faster than light signals and the price to be paid if we accept them: a very simple protocol Some physicists currently understand entanglement as transferring information instantaneously, yet not violating causality. Is this really a satisfactory explanation, or should be look for something better?
In particular, some people try to be explicit about a signalling between particles, but that does not seem consistent with relativity.
For example, if you have two particles, entangled and spatially separated, then the results of experiments can be predicted easily by a local hidden variable theory if the experimenters are forced to pick known (and equal) choices about what to measure (out of complementary variables, such as the $x$-component of spin and the $y$-component of spin).
So let's consider the situation where there are two labs (Alice's Lab to measure particle A, Bob's lab to measure particle B), and they are selecting (potentially) different things to measure.  You might postulate that if particle B were measured first, the result $a$ produced by particle A needs to be a function $a = a(X, Y, b)$, where $X$ is the type of measurement done on A, $Y$ the type done on B, and $b$ the result produced by B at the measurement $Y$.  Otherwise it is hard to agree with the correlations required.
Similarly, if particle A were measured first, then the result $b = b(X, Y, a)$. Otherwise it is hard to agree with the correlations required. But if you need the results of the other lab to generate consistent results here, how can the data from about $Y$ and $b$ be available to Alice's Lab?
Some people propose (and I argue against) that particles can transmit their results Faster than Light (FTL) to the other particle. In order to get the right results, it looks like the alleged faster than light protocol would go like:
We have two entangled particles A and B. A flies to a lab where the experimenter Alice works, while B flies to the lab of experimenter Bob.
The protocol is as follows:

*

*Alice measures her particle, A.


*Particle A transmits to particle B, by superluminal signals, the following information:
a) which measurement was done on A;
b) which result the particle A produced.


*The experimenter Bob measures his particle, B.


*Particle B transmits to particle A, by superluminal signals, the following information:
c) which measurement was done on B.
d) which result the particle B produced.
I don't get into details of which emitter/receiver A and B possess, and how many bits it takes to describe a type of experiment. I will just say that on the Earth the two experimenters appear to be done, at each trial, simultaneously.  By standard relativity arguments, if these events are spacelike separated different observers can disagree about which happened first, so there isn't a consistent story about which one happened first, and which sent information to the other.  The whole superluminal signalling program seems to be not enough to explain entanglement.  To provide the details, consider the following.
I'll focus on two travellers, Charlie and Dan. Charlie travels in a rocket in the direction of Alice, and Dan in a rocket in the direction of Bob. To an observer on Earth, their velocity is equal in absolute value. Therefore, by Charlie's clock, Alice measures in each trial before Bob, while by Dan's clock, Bob measures in each trial before Alice.
Now, let's ask Charlie what he can say about the above protocol: Particle A indeed sends to particle B, all the signals with info about the type of measurement done and the result. But, at the same time it sends to B all the signals with which type of experiment is done on B, and with which result. That, because superluminal signals that appear in one frame as sent from B to A, appear in another frame as sent from A to B.
But, no matter from whom to whom they are sent, the price to be paid if one accept this protocol is that at Alice's site and time-of-measurement, it's known which type of measurement Bob will choose, even before Bob will at all make a decision about which type of measurement to choose.
Thus, what should we do? Assume that a particle is endowed with prophecy about people's decision? Or, simply say that we don't yet understand properly how entanglements work, and as Danu formulated it better, to seek another explanation ?
Note: I would appreciate not to suggest changing the premises of the problem, e.g. not to suggest that the apparatus in one labs measures the particle in the other lab, or to disconsider relativity. And also not to propose me fuzzy ideas of the type (this or that) is done somehow, neither to suggest me what would be if some laws of the physics could be overridden. I would also appreciate to read the text before commenting it, and also to read the discussion that was done until now in the comments.
 A: Let me first say that I do not work in quantum foundations, really, so I might have a few misconceptions myself. I beg anyone to correct me, where I err and I will try to provide more references upon request.

After the question seems to have cleared up in chat, let me rewrite my answer:
You basically seem to ask: What if entanglement would allow superluminal communication in some way. And you propose a protocol. The protocol that you seem to advertise suggests active transfer of information. Since no experiment has ever seen anything like this, I guess we see this as a purely theoretical game.
Let's start with the obvious consequences: If we maintain our usual notions of special relativity, we could suddenly build a quantum cloner (in a way - yes I know there is criticism of this paper, but I don't see how this addresses the problem for us). This would leave us with a big problem, because the no-cloning principle is even more fundamental (in a sense) than superluminal communication, as it derives directly from the linearity of quantum mechanics itself. Already from here, there is no other possibility than to throw aside relativity and reconsider the classical notions of time and space. This is also the usual way I have seen superluminal communication being discussed in quantum information: As a way to show how this would lead to the usual violations of "no-cloning" or "Bell's telephone", etc.
This left aside, let's consider raw entanglement, which you actually want to talk about: Entanglement, as discussed in multiple posts, is nonclassical correlations. Now, when entanglement, as in your protocol, immediately sends information, it is no longer "just correlations", because it becomes active. And I don't see how to get around this "becoming active". At that moment, everything we understand about entanglement breaks apart - because it's no longer correlations, but it would be something completely different: an active link between particles. There are many posts here already describing how this is not possible (e.g. this one).

Now let's consider the protocol again, as the usual protocol involved in Bell's theorem. You claim that "Particle B transmits to particle A, by superluminal signals, the following information", which I interpret as "active transmission of information" - which would violate causality. You claim that this is transmission of information without violating causality and you claim (in chat) that:

All the quantum community researchers, distinguished professors, Nobel prize laureats, try to understand how the entanglement works.

While this is definitely an overstatement ("all" is just too strong), there are certainly many people trying to understand entanglement better. What they try to understand is which types of entanglement can be transformed into other types, how entanglement can provide help in certain protocols, etc. But hardly any one of them tries to understand what entanglement "is". The only answer you need to have in order to work with entanglement: entanglement is quantum-mechanically allowed and classically forbidden correlations. If you don't see
To answer your underlying question: "what is this sending of information with superluminal speed?". First of all, let's fix the terminology: "information", as it is usually defined, is something active: we can do something with it. This is something whose speed is limited by causality. Faster than light speed is nothing unusual: the phase velocity of a wave can be faster than light, even the group velocity of a wave packet can exceed the speed of light (anomalous dispersion case), but in these cases, the speed of information dispersion is still smaller than the speed of light. Since your "information" is something passive, it is no information.
This means that for special relativity, we do not (at all) have to care about something happening faster than light unless it is real information. So I think that your problem is not with superluminal information processing, but with the causal nonlocality of quantum mechanics and therefore, with the quantum state. I believe that you can get rid of the nonlocality by subscribing to an epistemic view instead of an ontic view, but this buys you other problems. This seems at the heart of the EPR-argument: you cannot have a fully ontological (realist) theory and complete locality. However, it is also something that most researchers I know don't care about, because it is too much philosophy for their taste. It is also not necessary, if one only wants a consistent theory to work with.
So I believe that your issue should be addressed in the debate on ontological and epistemological theories for quantum mechanics (see an overview about certain aspects of the discussion). Maybe you will find more satisfying answers there?
A: As you note, spacelike-separated (what you call "FTL") events can't be ordered unambiguously in time. That means such events can't be said to 'cause' one another in any reasonable sense.
One way around this is to abandon the idea of causation between events. If events don't really cause one another, but are actually just all equally caused by some sort of timeless agency - God, the universal wavefunction, whatever - then there is no problem with FTL. This doesn't seem to be the way things work, but it's certainly possible. Theories like this are said to be "super-deterministic", because they hold that counterfactuals - imagined statements different from the truth - not only don't obtain but are not even logically sensible.
However, this is not at all necessary because entanglement does not require one to believe in FTL signalling as long as the particles did not have definite properties in the first place. Measurements are correlated in kind of a weird way, but there's no possibility of noticing this until you are timelike-connected to both of them. In this, much more standard, interpretation, no inconsistency is ever observed. 
A: FTL signals are not only self-contradictory, as explained in my protocol, the nature doesn't use them. I repeat, they are not the way the nature works. (By the way, this is why we cannot lay our hands on such them, because the nature doesn't use them.)
For the rest, see my question and answer at (What stands behind the quantum nonlocality appearing in entanglements, and why Bell's inequalities are violated?)
