People often ask what it means for an electron to have spin 1/2.

Here is my attempt at an informal explanation.

It means that electrons (and most other elementary particles) are represented by wave functions or fields whose values are not given just by complex numbers (the “scalar” or “spin zero” case), but instead by complex vectors (of “internal” coordinates) which admit a finite dimensional representation of the rotation group. The action of a rotation on a state then corresponds to the usual change of position in space combined(*) with a reorientation of the “internal” coordinates.

It turns out (due to mathematics that I cannot usefully(*) insert here) that the possible results of measuring angular momentum corresponding to “internal” properties of a particle occur with increments of just half of those corresponding to measurements of the classical orbital angular momentum.

And the electron happens to be an example of the simplest kind of non-scalar field.

(*) – The crux of this can perhaps be inadequately explained by saying that the way the internal and external actions of the rotation group have to be related is such that one full rotation in the position space produces a sign reversal in the internal space and so to bring everything back to where it started actually requires two full rotations in the position space.