The following description is bogus. That it, this is not "really" what intrinsic angular momentum is all about; but it is possible to understand it in "common sense" terms, so we can use it as a mnemonic technique. Many concepts are introduced this sort of "cheating" until students get comfortable enough with them to define them rigourously. (The truth about spin, like much of QM, can never be made to seem sensible; it can only be gotten used to!)
Imagine a big fuzzy ball of mass spinning about an axis. While you're at it, imagine some electric charge sprinkled in, a certain amount of charge for every little bit of mass. (If you like, you can think of a cloud of particles, each of which has the same charge-to-mass ratio, all orbiting about a common axis.) Each little mass element contributes a bit of angular momentum and a proportional bit of magnetic moment, so that (summed over all the mass elements) and, as for a single particle, (constant). If the charge-to-mass ratio happens to be the same as for an electron, then (constant) , the Bohr magneton.
Now imagine that, like a figure skater pulling in her/his arms to spin faster, the little bits of charge and mass collapse together, making r smaller everywhere. To conserve angular momentum (which is always conserved!) the momentum p has to get bigger - the bits must spin faster. The relationship between L and is such that also remains constant as this happens.
Eventually the constituents can shrink down to a point spinning infinitely fast. Obviously we get into a bit of trouble here with both relativity and quantum mechanics; nevertheless, this is (sort of) how we think (privately) of an electron: although we have never been able to find any evidence for "bits" within an electron, we are able to rationalize its possession of an irreducible, intrinsic angular momentum (or "spin") in this way.
Such intrinsic angular momentum is a property of the particle itself as well as a dynamical variable that behaves just like orbital angular momentum. It is given a special label ( instead of ) just to emphasize its difference. Like , it is quantized - i.e. it only comes integer multiples of a fundamental quantum of intrinsic angular momentum - but (here comes the weird part!) that quantum can be either , as for , or !
In the following, s is the "spin quantum number" analogous to the "orbital quantum number" such that the spin angular momentum has a magnitude \ and a z component where is the chosen spin quantization axis. The magnetic quantum number for spin has only two possible values, spin "up" () and spin "down" (). This is the explanation of the Stern-Gerlach result for silver atoms: with no orbital angular momentum at all, the Ag atoms have a single "extra" electron whose spin determines their overall angular momentum and magnetic moment.