The way the book will describe it is from the perspective of group theory. If you assume that the laws of nature in a relativistic flat spacetime do not change under rotations, translations (in both space and time), and boosts (transformation between frames of reference of constant velocity), then these transformations define a Lie group called the Poincaré group. Representation theory then says that the representations of the Poincare group are characterized by two numbers. These two numbers are identified as a nonnegative mass and a spin. One interpretation is that these representations are indeed what "particles" are in Quantum Field Theory.
It is "intrinsic" because it has no known moving parts. If it had moving parts, and that motion had net angular momentum, we would refer to it in many contexts as "orbital angular momentum".
Spin should seem counterintuitive. There are no macroscopic objects that I am aware whose spin you can feel with your hands. Our research group has an object with 10^23 polarized spins (and negligible net magnetic moment) -- observing the angular momentum effects directly requires a sensitive modern torsion balance. https://arxiv.org/abs/0808.2673
Imagining the classic "figure skater experiment", the figure skater speeds up as she draws her arms inward, due to conservation of angular momentum. If she could shrink down to a point, her angular speed would tend to infinity. This is what I imagine happening to a black hole.
As a rule of thumb, that angular momentum corresponds to that at which the equator of the black hole is moving at the speed of light.
This has important consequences for the mergers of two high-spin black holes that have aligned spins. The merger is actually delayed for a little while, just before merger, as the binary dumps angular momentum through gravitational radiation.
One example: Neutrinos have just as much spin as electrons, but they have not yet been observed to have magnetic moments.
It is true that most magnets get some, if not all, of their magnetism from polarized electrons. We often state that the magnetism comes from "spins", as it is an effective shorthand, but it is not strictly correct. Some magnets get substantial amounts of magnetization from orbital magnetic moments of electrons (SmCo_5, for example).
Furthermore, one can create a magnetic field simply by moving an electric charge. From that perspective, a magnetic field is consequence of combining special-relativity with electrostatics.
We know a lot, but there is so much more left to learn.
This is where I get stuck. The particles are all excitations of a quantum field. So what does spin mean in that context?
I've not yet encountered an experimentally-testable explanation that goes deeper than that.