fromQuora: Isn’t a ‘probability wave’ simply a statistical function and not a real wave? Does it no more ‘collapse’ than me turning over a card and saying that the probability ‘wave’ of a particular deal has collapsed?

Well, as other answers have noted, the wave function of quantum mechanics is not a probability wave as its values are complex and it is only the squared amplitude that gives a probability density. But the process of “collapse” involved in a quantum measurement does involve something like your playing card analogy.

There are actually two stages in the measurement and observation process. One is the interaction with an incompletely known measurement apparatus which reduces or eliminates the prospects for future interference and basically turns the previously pure state of the isolated system (considered as a subsystem of the larger world) into a statistical mixture. And the second is the collapse of that statistical mixture by observation – with the result that, from the observer’s point of view, of the many possible alternatives only one is actually true.

And if this makes it seem to you that the “state” of the system actually depends on the observer then you are on the right track. (But it is nothing special about the “consciousness” of the observer that is relevant here. Almost any localized system could play the same role relative to the rest of the universe.)

Any configuration history of any physical system can be considered as “seeing” the rest of the universe in a “relative state” which “collapses” when the configuration history in question passes a point beyond which the configuration includes information about that particular measurement value.

Entanglement

Entanglement is just a word we can use to describe a situation where knowledge of some property of one object gives us information about some (possibly different) property of another.

The term is rarely used in the classical case, because we take it for granted. If we separate a pair of gloves for example and pack them up in identical boxes and then choose by a coin toss to send each in one of two opposite directions, then we are not surprised by the fact that if someone who knows how they started out but does not know the result of the coin toss opens one box and sees a left glove, he or she knows immediately that whoever opens the other box will see a right glove.

There is often similar classical chance-based uncertainty in our knowledge of quantum systems; but for such systems, even in the most precisely prepared “pure” states the knowledge of some properties makes it impossible for us to know others. This residual uncertainty is expressed by representing the state of the system by a “state vector” in a Hilbert space and the “mystery” of quantum entanglement is that the correlations between systems (like the gloves) that were once together but are now far apart can sometimes be greater than would be possible for any way of randomly assigning the properties at the outset.

The reason this extra correlation is sometimes considered “spooky action at a distance” is because the change of state vector (often called “collapse”) that occurs when we open one box seems to trigger a simultaneous collapse at the other box – and in a way that can change what the remote observer will see when looking at different properties from the one that obviously has to be opposite. At first sight it may seem that this effect might be used to send a signal where what the second observer sees might depend on what the first one chose to measure, but that turns out not to be the case.

Whether or not this bothers you may depend on whether you consider the state vector to be a property of the system itself or rather of the way it appears to a particular class of observers.