Spin is one of the defining properties of elementary particles, i.e. number that describes how an elementary particle behaves, much like electric charge and mass.
Possible values are half integer numbers: 0, 1/2, 1, 3/2, and so on.
The approach shown in this section: Section "Spin comes naturally when adding relativity to quantum mechanics" shows what the spin number actually means in general. As shown there, the spin number it is a direct consequence of having the laws of nature be Lorentz invariant. Different spin numbers are just different ways in which this can be achieved as per different Representation of the Lorentz group.
Video 1. "Quantum Mechanics 9a - Photon Spin and Schrodinger's Cat I by ViaScience (2013)" explains nicely how:
Video 1. Quantum Mechanics 9a - Photon Spin and Schrodinger's Cat I by ViaScience (2013) Source.
Video 2. Quantum Spin - Visualizing the physics and mathematics by Physics Videos by Eugene Khutoryansky (2016) Source.
Originally done with silver in 1921, but even clearer theoretically was the hydrogen reproduction in 1927 by T.E. Phipps and J.B. Taylor.
The hydrogen experiment was apparently harder to do and the result is less visible, TODO why: physics.stackexchange.com/questions/33021/why-silver-atoms-were-used-in-stern-gerlach-experiment
Needs an inhomogenous magnetic field to move the atoms up or down: magnetic dipole in an inhomogenous magnetic field. TODO how it is generated?
Video 1. Introduction to Spintronics by Aurélien Manchon (2020) Source.
Basic component in spintronics, used in both giant magnetoresistance
Video 1. What is spintronics and how is it useful? by SciToons (2019) Source. Gives a good 1 minute explanation of tunnel magnetoresistance.
Video 1. Introduction to Spintronics by Aurélien Manchon (2020) giant magnetoresistance section. Source.
Describes how giant magnetoresistance was used in magnetoresistive disk heads in the 90's providing a huge improvement in disk storage density over the pre-existing inductive sensors
Video 1. Introduction to Spintronics by Aurélien Manchon (2020) spin-transfer torque section. Source.
Describes how how spin-transfer torque was used in magnetoresistive RAM
Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles" has an amazing summary of the preceding chapters the spin value has a relation to the representations of the Lorentz group, which encodes the spacetime symmetry that each particle observes. These symmetries can be characterized by small integer numbers:
  • spin 0: representation
  • spin half: representation
  • spin 1: representation
As usual, we don't know why there aren't elementary particles with other spins, as we could construct them.
Leads to the Dirac equation.
Leads to the Proca equation.
Theorized for the graviton.
More interestingly, how is that implied by the Stern-Gerlach experiment?
physics.stackexchange.com/questions/266359/when-we-say-electron-spin-is-1-2-what-exactly-does-it-mean-1-2-of-what/266371#266371 suggests that half could either mean:
Initially a phenomenological guess to explain the periodic table. Later it was apparently proven properly with the spin-statistics theorem, physics.stackexchange.com/questions/360140/theoretical-proof-of-paulis-exclusion-principle.
And it was understood more and more that basically this is what prevents solids from collapsing into a single nucleus, not electrical repulsion: electron degeneracy pressure!
Bibliography:
Video 1. The Biggest Ideas in the Universe | 17. Matter by Sean Carroll (2020) Source.
Video "The Biggest Ideas in the Universe | 17. Matter by Sean Carroll (2020)" at youtu.be/dQWn9NzvX4s?t=3707 says that no one has ever been able to come up with an intuitive reason for the proof.

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