The mention of “spin” of a particle is one that immediately triggers an intuitive, simple visualization in the minds of nearly everyone: perhaps, a ball rotating with some angular velocity about an internal axis. This is certainly what was visualized by some of those originally peering into the work of the young Wolfgang Pauli (he of the Exclusion Principle fame), but it is a short-lived analogy: the original name of the phenomena was a “two-valued quantum degree of freedom”, and it behaves only as such.
In a simple sense, elementary particles do experience a sort of spin — that is, an intrinsic angular momentum about their own respective axes. However, the analogy of the spinning top comes to a halt when another characteristic of spin comes into play — it can only exist in one of two possible orientations at any given time. This is not a quality that is exhibited by any spinning top. This is why physicists deem spin a “non-classical” degree of freedom — another way of stating that it is a phenomena not described by any macroscopic physical analogy.
Another peculiar property of particle spin is that it can either be 0, or any half-integer value (usually implied to be a multiple of ħ, the reduced Planck constant). Stephen Hawking made a clever analogy is made to demonstrate the counter-intuitiveness of particle spin: imagine a deck of playing cards. A spin-0 particle behaves like a point, and appears similar in every orientation. A spin-1 particle behaves like an ace: a rotation of 360˚ brings it to its original orientation. A spin-2 particle behaves like a face card: one half-rotation brings it to its original orientation. The peculiar point is that a spin-1/2 particle, for example, does not behave in this way: it takes two complete revolutions to return to its original position.
A final remark on the concept of spin and its importance: the Standard Model of particle physics is fundamentally interconnected with spin. The way we categorize elementary particles — into fermions (“matter” particles) and bosons (“force” particles) — is based on the fact that the former are spin-1/2 particles, and that the latter are of spin-1.
The intrinsic symmetry and anti-symmetry present with bosons and fermions, respectively, are what allow for (somewhat) stable particles to form — protons, neutrons, etc. — and without this completely outlandish and truly bizarre phenomenon, the known universe would not exist as we know it today. Spin is but one example of how ill-equipped humans are to intuitively comprehend the fundamental processes that govern our universe, and an example of how striking it is that we remain unfazed while making deeper-penetrating progress.
I know I’ve already done a post on spin, forgive me! But this is a great explanation!