Quantum Entanglement, connected or just in tune? I read a few of the questions suggested to be similar, but I didn't see what I am looking for. Please feel free to point me in the right direction if I just missed it... (there are quite a few).      
I am in my mid 30s and never went to college. My understanding is largely from various YouTube sources.  
There is one thing I don't understand about entanglement and I can't find any videos that explain it.       
If I spin a coin on a table, shoot it with a gun dead center and the coin breaks perfectly into two smaller coins... and the split pushes the coins into spinning in opposite directions of each other... and this was all on a big magic friction-less table that allowed them to spin away from each other until they were a mile away from each other... I could look at one and see that it was spinning clockwise and instantly know that the other coin is spinning counter clockwise without those coins having to communicate faster than light.           
I know I'm just some pleb stuck thinking from an intuitive state of mind... but I'm open to expanding that... I just don't understand why entanglement isn't just two objects being in sync because they were created in sync or were synced up at some point. Can anyone explain why this isn't the case? 
 A: You are asking a good question. 
There is a difference between quantum entanglement and classical correlation (like the example of the spinning coin you just gave). This difference however is quite subtle, and it requires some good thought to explain and understand, as it involves some probability theory. The keyword is Bell's Theorem.
Indeed, most popular explanations of entanglement are quite limited and do not get to the point. And that is why you, and many of us asked questions like yours. 
If you want to understand, I'd suggest reading David Mermin's classic paper Is the moon there when nobody looks. You will understand the subtle difference between quantum entanglement different and classical correlations -- as well as why it is actually a profound difference.
You can also look at this minutephysics video on the Bell's inequalities.
A: In you example with the coin there is some set of measurable quantities like position, momentum, angular momentum etc. On the scale of everyday life for most purposes you can measure all of those quantities to a higher accuracy than is needed for most applications without affecting the results of similar measurements on other quantities.
Now, suppose an atom that emits a pair of spin entangled electrons in a singlet state. Spin is a kind of angular momentum and it is quantised and comes only in quantities +1 or -1 in suitable units. You can measure the spin along multiple directions like x,y,z or a direction in between those directions. Now suppose you measure each electron in the entangled pair in either the x or y direction chosen randomly. For each electron the probability of getting +1 is 1/2 and the probability of -1 is 1/2. If you measure both electrons in the x direction and compare the results, they will be different: if one electron has +1 the other will have -1. The same is true if you measure both electrons in the y direction. But if you measure one in the x direction and the other in the y direction, then the results will match with probability 1/2. So if you get the result +1 for x spin on one of the pair and compare the result with a measurement of the y spin on the other electron you will find the other electron has +1 with probability 1/2 and -1 with probability 1/2. So the probability of a match depends on both measurements and you can do mathematical calculations  to show that classical physics can't match that result.
The standard explanation is either that you don't try to explain this result at all, or you say the electrons somehow interacted non-locally. 
There is an explanation that most physicists reject or don't know about for reasons that don't make a lot of sense to me. The gist of the explanation is that both results happen for each electron: there are two versions of the electron. When outside systems such as measuring instruments interact with the electron there are two versions of the measuring instrument after the interaction and there are also two versions of all the systems that receive that information about the electron, including you. The correlations between the results are only established when systems carrying the results interact not when the electrons are measured and they are established entirely locally without any action at a distance. The reason you don't see these other versions of yourself and everything around you is that they are divided into layers that don't interact with one another to a good approximation because quantum mechanics forbids them from exchanging information. The approximation isn't perfect for reasons, e.g. - all the electrons in the atoms in your exist in multiple versions but the scale you would have to measure to see this effect is smaller than that of everyday life. See "The Fabric of Reality" and "The Beginning of Infinity" by David Deutsch and the papers below:
https://arxiv.org/abs/quant-ph/0003146
https://arxiv.org/abs/quant-ph/9906007
https://arxiv.org/abs/1109.6223
A: Your coin example is analogous to only one specific state of entanglement. That is referred to as Bell's state. This is a state of perfect anti-correlation. I will only talk about this state. Taking an example of spin - if the two particles of entangled pair are measured along same axis, then they will be always found to be opposite spin. Another way to say "always opposite spin" is perfectly anti correlated.
Perfect anti correlation can be easily explained in terms of your example. You do not even need to shoot a coin, you may just separate a pair of shoes, or socks, or gloves! Perfect anti correlation is a consequence of conservation laws. In case of spin, if one particle gets clockwise spin along X axis, then the other has to get anti-clock wise spin along same axis, in order to conserve angular momentum.
Next level of complexity is that you measure them along any axis, and you will see they have opposite spin. This is complex, but still explainable in terms of conservation laws. Specifically because, there is no measurement at quantum level, it is an alignment which is called "measurement".
Above two, in isolation, can be explained in terms of local hidden variables. Because, the above two involve only one pair at a time for checking the anti correlation.
But there is another level of complexity - Video suggested by Aaron Stevens is one of the best.
Third level of complexity is the one that baffles most people. It is statistical correlation. By definition, statistical correlation, has to involve many pairs.
Suppose you measure one particle of the pair along X axis and the other particle along an axis that is at theta degrees to X axis. There are only two possibilities - 1. they show same spin, 2. they show opposite spins. If they show same spin, let us say they are correlated.
Now you repeat same as above on 2nd, 3rd, .... n pairs. For a very large n, the number of correlated pairs is predicted by quantum mechanics as a formula which I will not write here.
All experiments show that the QM formula is correct.
Then there is a Bell song that you will hear a lot, almost from everyone. If you pay attention to the video, Bell considered all possible outcomes equally likely and came up with an inequality, which obviously, is violated in all experiments.
Bell's theorem as mathematics is just fine. But the issue with applying Bell's inequality to statistical correlations of entangled particles is - All the possible outcomes are not necessarily equally likely.
Even in a process that is as simple as hitting a mountain continuously with a hammer, your past hits have an impact on outcome of subsequent hits. 
So having generated and aligned numerous pairs, we can not say the possible outcomes for the next pair are equally likely. And this fact renders application of Bell's inequality to entanglement - void.
So finally, entanglement is baffling, but has been mathematically formulated pretty well. Only thing missing is some realistic explanation which I think will turn out to be much simpler than it seems. Nature is simple at the core.
A: The two halves of the coin split.  You observe one, either through your left eye or your right eye.  I observe the other, either through my left eye or my right eye.  If we both look through our left eyes, the coins reliably spin the same direction.  Ditto if you look through your left eye and I look through my right.  Ditto if you look through your right eye and I look through my left.
But whenever we both look through our right eyes, the coins reliably spin in opposite directions.
That's entanglement.  (More precisely, it's an extreme violation of Bell's Theorem, even more extreme than entanglement allows.)  Your gun story cannot account for these statistics, and (for the same reasons) cannot account for the  actual statistics predicted by quantum mechanics (which are also the actual statistics seen in experiments.)
Quantum entanglement is not classical correlation.  As for why --- as I said in my comment, you might as well ask why a ray of sunshine is not the same thing as a squirrel.  They're not the same thing.  Why should they be?
A: The reason is that quantum phenomena exist. That is, in every other situation but an entangled partner, it should be impossible to know the quantum state. 
Because it is considered proven that quantum outcomes are truly unknowable and probabalistic, not governed by any hidden variable, accounting for this seems to imply an instantenous transfer of information  between the entangled pair, when the wavefunction of one of them is collapsed. Which Einstein certainly thought impossible. The consensus is that he was wrong, in this specific type of phenomena. Although, it is very likely an area for new physics to resolve.  
