What are Einstein’s “spooky actions”?
Quantum mechanics tells us that we can never know what state an object/particle is in until we make a direct measurement. Until then, the object exists in a superposition of states, and we can only know the probability that it is in a given state at a given time. Making a measurement disturbs the system, and causes those probabilities to reduce to a single value. This is often referred to as collapsing the wave function,
Einstein was uncomfortable with the probabilistic nature of quantum mechanics. He felt that physical objects should have definite properties regardless of whether or not they were being measured. He is famously quoted as asking, "do you really believe the moon is not there when you are not looking at it?”
He used the phrase "spooky action at a distance" to refer to the foundational notion of QM that making a measurement of one object can somehow directly affect the measurement of another object in a different region of space, with the two objects located an arbitrary distance apart. This notion is called quantum entanglement, and Einstein didn't like it.
Suppose that we have two spheres, one red and one blue. We put each of the spheres in a box, and then we mix the boxes up until it is impossible for us to know which sphere is in which box. Intuition tells us that even if we don't know which sphere is in which box, one of them must be red and the sphere which is not red must be blue, i.e. the first box contains a red sphere and the second box contains a blue sphere, or the first box contains a blue sphere and the second box contains a red sphere. On the other hand, quantum mechanics tells us that until we open the boxes, the spheres exist in a superposition of red and blue, i.e. they are both red and they are both blue.
When we open one of the boxes and see the blue sphere, we know the other box must contain the red sphere. We know this without opening the other box. We could keep the second box closed for the remainder of time, and it would still always be known that the second box contains the red sphere. Knowing something about one of the objects (that it is blue) gave us information about the second object (that it is red), without having to make a direct observation of the second object. Therefore, we say that these two objects are entangled.
This would be true regardless of whether or not quantum mechanics is correct. Even if the objects held definite states the whole time, looking at one would give us information about the other. But oddly enough, experimentation so far has confirmed the quantum mechanical interpretation every single time.
Quantum entanglement tells us that when we make an observation of one of the spheres and see that it is red, that object must somehow "communicate" with the other object and tell it which state it needs to be in. In this case, when we see the red sphere, the red sphere must tell the sphere in the other box that it needs to be blue. When we open one box and see the red sphere, the wave function of that sphere collapses, but the wave function of the second sphere collapses as well. If not, we might have the situation where both objects are red or both objects are blue, which we know would be impossible.
Einstein was strongly opposed to this idea. In 1935 he published a paper in which he attempted to disprove quantum theory. This is famously known as the EPR paper, after the three authors (Einstein, Podolsky, and Rosen). The thought experiment proposed that in order for quantum mechanics to be correct, it must mean that information can travel faster than the speed of light, which directly violates Einstein's theory of relativity. As it turns out, Einstein was incorrect; quantum entanglement does not result in a paradox. If you would like more information about the EPR paradox, feel free to send me a message! There are also a lot of good resources one can find on the internet.