electron spin
It is tempting to picture an electron as a tiny spinning top, twirling on its axis like the Earth. That image gives the idea its name, and it explains why an electron acts like a miniature magnet with a north and south pole. But the picture is only a crutch: electron spin is a genuinely quantum property with no true spinning ball behind it.
Electron spin is an intrinsic form of angular momentum carried by every electron, independent of any motion through space. It is quantised: an electron can only be 'spin up' or 'spin down', conventionally +½ or −½. This spin gives the electron a magnetic moment, so atoms respond to magnetic fields, and it supplies the fourth quantum number that completes an electron's description.
Spin matters enormously: it is what the Pauli exclusion principle counts when it limits each orbital to two electrons, and it underlies magnetism, the splitting of spectral lines, and the operation of MRI. The honest caveat is that nothing is literally rotating — spin has no classical analogue, and treating it as a physical spin breaks down if pushed too far.
In the famous Stern–Gerlach experiment, a beam of silver atoms passed through an uneven magnetic field splits into exactly two beams — never a smear in between. That clean two-way split is direct evidence that the outer electron's spin can take only two values, up or down.
Spin is quantum angular momentum with only two values — up or down.
Two paired electrons in one orbital have opposite spins, so their magnetic effects cancel; a lone unpaired electron leaves the atom magnetic (paramagnetic).