Brian Bi's answer to Does the spin of subatomic particles actually involve angular momentum (i.e., is the particle actually *spinning*)?

Yes, it involves angular momentum. That spin angular momentum is true angular momentum can be demonstrated by the Einstein–de Haas effect. Angular momentum is seen to be conserved if and only if we account for the change in angular momentum resulting from the alignment of electron spins within the ferromagnetic rod.

But no, it doesn't involve spinning! Spinning occurs when a body revolves around an axis that passes through it; every particle in the body travels in a circular path around the axis. For a body consisting of a single elementary particle, there can be no spinning of this sort because there are no smaller particles inside the electron. Either the electron moves in a circular path, in which case it isn't spinning, or it stays in one spot, in which case it also isn't spinning because nothing is moving.

Erik Anson raises the question of what it means for a particle to have intrinsic angular momentum, which can also be phrased as the question: for what reason do we say that electrons have intrinsic angular momentum? If you say that electrons have no intrinsic angular momentum then how can I prove you wrong? Is angular momentum something you can read off a meter? (Answer: probably not.) If not, then what's the theoretical definition of angular momentum that allows us to conclude that it is carried intrinsically by some particles? The rest of my answer will address this.

The distinguishing characteristic of particles that have spin is that they have an extra property attached to them other than position or momentum which is affected by rotation, or, in more technical terms, "an internal degree of freedom acted on by the Lorentz group". Sort of like a tiny arrow attached to the particle; an intrinsic orientation; although more complicated than that. A particle without such an internal degree of freedom is analogous to a sphere: it has no directionality. Some particles do have such an internal degree of freedom, which is like having a lopsided "shape" that isn't the same in all directions. This shape has nothing to do with spatial extent; it's just an analogy.

Photons and electrons do have such an internal degree of freedom; they are "lopsided". When you describe a photon or an electron, it's not enough to just give its position or momentum; you also have to supply some extra numbers to describe their orientation. There are some particles that don't have such an internal degree of freedom; you only need to know the position or momentum and you know all there is to know. These are the scalar, or spinless particles. The Higgs boson is a scalar particle.

Now we have to introduce the next ingredient, which is Noether's theorem. Noether's theorem relates symmetries to conservation laws. One particular symmetry of the universe is rotational symmetry: an isolated system behaves the same way no matter what its orientation is; it behaves the same way as another isolated, rotated copy of itself. When you rotate a system, the electrons and photons inside it have their positions moved around your axis, their momenta rotate too, and their spins also rotate, but despite the fact that everything changes, the new system still behaves the same way. Because there is such a symmetry, there is also a conservation law, a law that states that a certain quantity is conserved because there is a symmetry, and gives a formula for that quantity. For a detailed introduction to Noether's theorem, see this answer and links therein. The formula is given there:

[math]j = X(c_0) \cdot \frac{\partial \mathcal{L}}{\partial \nabla c} - f[/math]

You don't need to fully understand this formula, but here's what you should understand. The requirement is that there is a symmetry, so that when you change the system somehow, the laws of physics stay the same. [math]X[/math] tells you how the system changes, the same way the tangent vector of a curve tells you how the position of a particle travelling along that curve changes. The first term in [math]j[/math] is proportional to [math]X[/math].

A particle's position certainly changes when it rotates (unless it's located along the axis of rotation). That's true no matter what kind of particle it is. So when you do a rotation, [math]X[/math] is like a circumferential tangent vector that tells the particle which direction the rotation carries it in. So you plug that [math]X[/math] into the formula and you calculate a value for [math]j[/math], and you get a quantity called orbital angular momentum. This is the "classical" type of angular momentum, the "L equals r cross p" type of angular momentum, which bodies have when they revolve or circulate around an axis.

But since some particles have an additional property that changes under a rotation, you also have to include the way that internal degree of freedom changes in [math]X[/math]. For a particle with spin, this will be nonzero, since that internal degree of freedom does change. For a scalar particle, it's zero since we already accounted for all the change that happens to it under a rotation; there's no internal degree of freedom. The additional contribution to [math]X[/math] for particles with spin is responsible an additional contribution to [math]j[/math], and that contribution is the spin angular momentum. To get the actual conserved [math]j[/math], you have to include all effects of the rotation in [math]X[/math] and hence [math]j[/math]. If you don't do so, the quantity you calculate won't be conserved since it'll be missing terms. So the total angular momentum is the sum of orbital and spin angular momentum. A scalar particle has only orbital angular momentum.

The spin of the electron explains why electrons can have magnetic dipole moments. The electron is lopsided so it's possible that it has a magnetic field that comes out of one end of it and goes into the other end. The spin of the photon explains why photons can have polarization. A spinless particle can have neither of these things; it cannot have any observable properties that are directional, except for momentum.

tl;dr: some particles have an internal degree of freedom acted upon by rotations, and others do not; and those in the former category have, by definition of angular momentum as a Noether charge, a contribution to their angular momentum from the way the internal degree of freedom is acted upon by rotations, while those in the latter have no such contribution; and particles in the former category are said to have spin while particles in the latter category are said to be spinless or to have spin zero.