A single electron has spin, and measured along any axis it gives one of two results, up or down. Its state can be up, down, or a superposition of the two, \alpha|{\uparrow}\rangle + \beta|{\downarrow}\rangle, in which |\alpha|^2 and |\beta|^2 are the probabilities of the two results.
Two electrons together have four basic states, |{\uparrow\uparrow}\rangle, |{\uparrow\downarrow}\rangle, |{\downarrow\uparrow}\rangle and |{\downarrow\downarrow}\rangle. If each electron has a state of its own, the state of the pair is the product of the two, and each electron can be described without the other.
Some states of the pair are not products. The singlet, |\psi\rangle = \dfrac{1}{\sqrt{2}}\left(|{\uparrow\downarrow}\rangle - |{\downarrow\uparrow}\rangle\right), is one of them. Multiplying out (\alpha|{\uparrow}\rangle + \beta|{\downarrow}\rangle)(\gamma|{\uparrow}\rangle + \delta|{\downarrow}\rangle) gives \alpha\gamma|{\uparrow\uparrow}\rangle + \alpha\delta|{\uparrow\downarrow}\rangle + \beta\gamma|{\downarrow\uparrow}\rangle + \beta\delta|{\downarrow\downarrow}\rangle, and no choice of the four numbers removes the first and last terms while keeping the two in the middle.
A state of this kind is called entangled, a word Schrödinger introduced in 1935. In the singlet the total spin is zero, and neither electron has a spin direction of its own. The pair stays in this state while the two particles move apart, as long as nothing measures or disturbs either of them.
Let the two particles move apart to two experimenters, usually called Alice and Bob, who each measure the spin along the z axis. Alice gets up or down with probability 1/2 each, and so does Bob.
When they compare their records afterward, they find that every pair gave opposite results. Whenever Alice got up, Bob got down, and the other way round. The singlet has total spin zero, and the two results add up to zero for every pair.
The singlet has the same form along every axis, so the same holds if both turn their magnets to x or to any other common direction. It also holds however far apart the two measurements are made, and whichever of them is made first.
Opposite results along one axis could also come from pairs prepared in advance with opposite spins. The difference between that and the singlet appears only when the two sides measure along different axes, as the following scenes show.
Bob's own record, looked at without Alice's, is a random list of ups and downs with about half of each. This does not change when Alice turns her magnet from z to x, or stops measuring altogether.
The reason is in the state itself. For the singlet, the probability of each result on Bob's side is 1/2 along any axis, whatever is done on Alice's side. Alice's choice changes how her results are related to Bob's, but not the numbers Bob sees.
So entanglement cannot be used to send a message. Bob cannot tell from his own results whether Alice measured, or along which axis. The correlation appears only when the two records are brought together by ordinary means, such as a phone call or a letter, and these travel no faster than light.
The rule that the statistics on one side cannot depend on the choices made on the other side is called no-signaling. Every entangled state obeys it, which is why the correlations do not conflict with relativity even though they do not weaken with distance.
Now let Bob turn his magnet by an angle \theta from Alice's. Quantum mechanics gives the probability that the two results are the same, both up or both down, as P = \sin^2(\theta/2), and the probability that they are opposite as \cos^2(\theta/2).
At \theta = 0 the results are never the same, as in the second scene. At \theta = 90^\circ they are the same half of the time, so the two records show no correlation at all. At \theta = 180^\circ, when Bob's magnet points the opposite way, the results are always the same.
It is convenient to give each result the value +1 or -1 and to average the product of the two values over many pairs. This average, the correlation, is E(\theta) = -\cos\theta for the singlet. It is -1 when the results are opposite every time, 0 when they are unrelated and +1 when they are the same every time.
The curve \sin^2(\theta/2) is flat at \theta = 0. Turning Bob's magnet by a few degrees changes the results very little at first, and then more and more. A model in which each particle carries answers fixed in advance cannot give this shape for all angles at once, and Bell's inequality states exactly how.
Experiments usually use photons instead of electrons. The polarization of a photon takes the place of spin: a polarizer either lets the photon through or stops it, two results like up and down. Entangled pairs are made by sending laser light into a crystal in which a small fraction of the photons each split into two photons of lower energy.
For photons the angles are halved. Two polarizers at 45^\circ to each other correspond to two magnets at 90^\circ, and for the usual photon pair the results agree with probability \cos^2\varphi when the polarizers are at an angle \varphi.
In 2017 the Chinese satellite Micius sent entangled photon pairs from orbit to two ground stations, Delingha and Lijiang, 1{,}203 km apart. The polarizations measured at the two stations showed the correlations that quantum mechanics predicts, although the two measurements were made too close together in time for any signal at the speed of light to pass between them.
Results like these do not depend on how far apart the particles are. They also cannot come from a list of answers given to each photon before it leaves the source, and the reason that is impossible is Bell's inequality.