I read about quantum spin a while ago. If it’s not spinning, what is it doing?
The word “spin” is used for historical reasons, but that property of quantum particles has nothing to the normal meaning of the word. It would have been better if we called it “intrinsic angular momentum” from the beginning, but it’s too late now - physicists used the word “spin” a century ago and it has stuck.
As an analogy (which probably has some flaws) you can try to exchange the following concepts:
spinning <-> movement
angular momentum <-> energy
Then, it is true that when a particle is spinning, it has some angular momentum. Similarly, when a particle moves with some velocity is has some energy associated with that movement.
Now, this doesn't mean that any energy that the particle has is due to its movement, nor all the angular momentum is due to the spinning of the particle.
There is an intrinsic energy that the particle has only because it exists, is not due to any movement, we call it mass.
Similarly, there is an intrinsic angular momentum that the particle has only because it exists, not due to any spinning or anything else. We call it spin.
The analogy given by @Gaussian97 points to spin and (invariant) mass being somewhat analogous.
But one could ask why choose energy and not (linear) momentum as comparison. After all, if there exists an intrinsic angular momentum for a massive particle without an angular movement, couldn't there also be an intrinsic linear momentum for a massive particle without linear movement?
Well, if we talk about the 3-momentum of classical mechanics, the answer is of course no. No linear movement means using the rest frame of the particle as reference for the description and there, the 3-momentum is zero. But even in this frame, the magnitude of the 4-momentum of relativity is equal to the invariant mass of the particle. So in this sense, mass plays the role of an intrinsic linear momentum which strengthens the analogy between spin and mass.
I think it's still a quite shaky analogy but there is a rigorous way in which spin and mass are analogous: both play similar roles in the application of representation theory to spacetime symmetries in QM. It's usually done within the relativistic context but does already work in non-relativistic QM.