Spin is intrinsic angular momentum: it obeys angular-momentum algebra but is not caused by a particle literally rotating as a classical extended object.

A spin-$1/2$ system has two possible outcomes for a component measurement. Along the $z$ axis, $$S_z|+z\rangle=\frac{\hbar}{2}|+z\rangle,\qquad S_z|-z\rangle=-\frac{\hbar}{2}|-z\rangle.$$ The spin operators can be written using Pauli matrices: $$S_i=\frac{\hbar}{2}\sigma_i.$$

The Stern-Gerlach experiment sends particles with magnetic moment through an inhomogeneous magnetic field. Instead of a continuous spread expected from arbitrary classical orientations, the beam splits into discrete components corresponding to quantized spin projection.

A state prepared as $|+z\rangle$ is not an eigenstate of $S_x$. In the $x$ basis, $$|+z\rangle=\frac{1}{\sqrt2}\left(|+x\rangle+|-x\rangle\right),$$ so measuring $S_x$ gives $+\hbar/2$ or $-\hbar/2$ with equal probability. A subsequent $S_z$ measurement no longer returns $+\hbar/2$ with certainty because the intermediate measurement changed the state.

Spin-$1/2$ systems provide the simplest nontrivial quantum two-state space. They appear in electrons, protons, neutrons, magnetic resonance, qubits and the construction of many-particle quantum states.