The concept of spin is ground-zero in relativistic quantum mechanics -- further removed from intuitive macro behavior than other behaviors we consider quantum.
If you took undergraduate quantum mechanics, at some point you were introduced to the concept of "spin." If you're like me, you left that class feeling you were shown a magic trick but not how it worked. Don’t worry; you are not alone. The reason you weren't told more is not that a better explanation was left for graduate quantum mechanics. You weren't told more because there isn't a lot more to know.
Of course, there is the magnetic dipole moment. Shoot a beam of electrons through a magnetic field with a gradient, and the beam will bend thanks to the influence of the field on a moving charge. Furthermore, it will also split into two sub-beams -- one containing "spin-up" electrons and the other containing "spin-down" electrons. An oriented magnetic dipole moment in the macro world corresponds to circulating current. Hence, the image of a spinning electron, presumably with a non-uniformly distributed charge that gives rise to the magnetic moment.
But if you do the math with some sort of estimate of electron size, the charge has to be spinning at many times the speed of light, which is not possible. Actually, it's not clear the electron actually has a size, so what is spinning anyway? Not that the spin isn't real -- you can induce electron spin transitions in an atom with polarized light, and photon spin (circular polarization) is real. But the intuitive explanation for the dipole moment is completely wrong, and we still don't have any concept of just what is spinning.
When we dig deeper, we learn that electrons -- like most familiar particles -- have a spin of ½. I'm sorry, ½ of what? Physicists, when confused, look at symmetries to understand behavior. A common example is rotational symmetry. A simple square rotated through 90 degrees looks identical to the un-rotated square. The same applies to 180 degree, 270 degree, and 360 degree rotations. Virtually everything looks the same if you rotate it through 360 degrees -- it's like no rotation at all.
But not spins of ½. Those you have to rotate through 720 degrees to get back to where you started, which is where the ½ comes from -- a spin of 2 is rotationally symmetric at 180 degrees, a spin of 1 at 360 degrees, and a spin of ½ at 720 degrees.
A neat trick to demonstrate this type of rotational symmetry (and a way to celebrate your inner geek at parties) is something called the plate-trick. Lay a plate flat on your hand, and then rotate it under your arm, over your head, and back down to the original position, all the time keeping the plate flat. If you watch carefully, you will notice that you rotated the plate through 720 degrees -- the same kind of rotational symmetry as spin ½. So we've discovered a real-world, albeit contrived, example.
But what does this mean for electrons? If you rotate polarized electrons through 360 degrees, why isn't that the same as no rotation? In fact, it is almost the same, but the sign of the wave function reverses -- you have to rotate through 720 degrees to get back to exactly the same wave function. The sign reversal has no observable effect if you're just looking at the intensity of the beam, but it can be seen in the circular polarization of light emitted from atoms excited by the electrons. Here again, spin behavior is, at best, on the fringes of intuition.
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