Spin is a pretty abstract thing, not having a really good analogue in classical physics. The best way to establish the quantum-mechanical rules are symmetry principles, and the very first symmetry principles to be used to build a concrete quantum theoretical model for, e.g., elementary particles, are those arising from the symmetry of the underlying space-time model.
In non-relativistic (Newtonian) or special-relativistic physics space is represented by a three-dimensional Eucildean affine manifold and thus is homogeneous and isotropic. Thus translations and rotations are symmetries of space and thus must be represented in quantum theory. In addition there's also the time-translation and boost symmetry. Due to a famous analysis by Wigner this symmetry group of classical physics must be represented by a unitary ray representation of this group on Hilbert space. Since the group is continuous (more precisely that's the case for the proper orthochronous Galilei transformations and the proper orthochronous Poincare transformations), this is equivalent to a unitary representation of a central extension of the universal covering group.
In non-relativistic physics the central extension is given by the Galilei group with the rotation group SO(3) substituted with its covering group SU(2) and the mass operator as central charge. The spatial translation subgroup gives rise to momentum \vec{p} as an observable and any momenum-eigenstate can be reached from the zero-momentum eigenstate by a Galilei boost, represented by the corresponding unitary operator. Thus one has only to deal with the subgroup which leaves the zero-momentum eigenstate invariant to get all possible representations. The group which leaves the zero-momentum eigenstate invariant obviously is the rotation group, represented by SU(2).
This means that the intrinsic structure of a quantum system with vanishing momentum is characterized by its mass (i.e., the value of the central charge of the representation) and the representation of SU(2) according to which the zero-momentum states are transforming under rotations. The unitary representations of SU(2) are well-known and derived in any good textbook of quantum theory, leading to the representations of angular momentum, which are nothing else than the generators for rotations. Thus "spin" tells us, how the zero-momentum eigenstates of the system transform under rotations.
In non-relativistic physics one can uniquely separate the total angular momentum in a spin and an orbital angular momentum part, because the boosts, which generate any momentum eigenstate out of the zero-momentum eigenstates built an Abelian subgroup of the Galilei group, not mixing with the rotations.
In relativistic physics the Poincare group is somwhat simpler than the Galilei group, and all ray representations are simply induced by unitary representations of the covering group, which substitutes the special orthochronous Lorentz group \mathrm{SO}(1,3)^{\uparrow} by \mathrm{SL}(2,\mathbb{C}). The very same analysis of these representations leads to very similar notions for massive states (mass is given as the Casimir operator of four-momentum p^2=m^2, i.e., it's not a central charge as in Newtonian physics). Only it's not possible to uniquely split the total angular momentum into spin and orbital part, because the boosts themselves build no subgroup but only boosts together with rotations. For massless states it's a bit more complicated: There you have helicity instead of spin as the corresponding intrinsic quantum number of the system.
For details, see my qft manuscript
http://fias.uni-frankfurt.de/~hees/publ/hqm.pdf
or Weinberg, Quantum Theory of Fields, Volume 1.
So, spin is a rather abstract notion. The idea of imaging elementary particles with spin as rotating little bullets is dangerous. This picture is even dangerous for spinless particles since it's only a classical approximation which holds not true if quantum theory is really important. The best picture we have about them is quantum field theory and nothing else. We have to live with this rather abstract picture since our senses are simply not trained for the microscopic world, where quantum theory has to be applied. Except the fact that the matter among us is pretty stable, we don't experience much quantum-theoretical behavior of the macroscopic objects we deal with, so that we cannot have intuitive notions about it from everyday experience.
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