Could someone please attempt to explain the the angular momentum quantum numbers?
My chemistry book does a slapdash job of it, and, well, I'm confused here!

any quantum degree of freedom that is restricted to some compact set will exhibit a discrete spectrum, or a discrete set of allowable values. for the case of angular momentum, the compact set in question is the rotation group.

in simple terms, angular momentum comes in discrete values because angles can only run from 0 to 360, and cannot run to infinity.

if you put a particle in a box, you will see the same thing happen to its momentum, if the box has a finite size. if the box is infinite, then the particles momentum is not discrete.