Short answers, pulled from the story.
The beam of silver atoms split into two distinct spots on a detector screen. This result showed that the atoms possessed two possible discrete angular momenta despite having no orbital angular momentum.
These mechanical pictures failed because the required rotation speed would exceed the speed of light at the electron's hypothetical radius. Wolfgang Pauli initially rejected any idea that this new degree of freedom related to physical rotation and called it classically non-describable two-valuedness instead.
A full 720-degree rotation is necessary to restore the exact initial condition for a spin-1/2 particle. Rotating such a particle by 360 degrees does not return it to its original quantum state but flips its phase sign.
Wolfgang Pauli published a proof in 1940 connecting particle spin magnitude directly to quantum statistics. The theorem relies on both quantum mechanics and the theory of special relativity to establish its validity.
The electron possesses a nonzero magnetic moment determined experimentally to have a value near 2. This measurement validates predictions made by the Dirac equation regarding electromagnetic properties.