At a macroscopic level the effects of quantized spin are best observed in the setting of the Stern-Gerlach experiment. Classically, we expect that the magnetic moment associated with a charge is related to its angular momentum. Since the electron has a non-zero angular momentum, we expect that it will experience a force in the presence of an inhomogenous magnetic field that depends on the orientation of the electron's hypothetical "axis of rotation". In practice, the inhomogeneous magnetic field is directed from a narrow north pole to a "spread-out" south pole. If the angular momentum of the electron behaved classically, then the "spin" of the electrons would be distributed uniformly from a thermal source (in order to maximize entropy), and when the electron is "shot" through the magnetic field to be detected on a screen, we would expect a roughly even distribution of position measurements. Instead, however, what we observe is two cleanly separated populations of electrons: one family which deflects upwards, and another which deflects downwards. This is inconsistent with the classical picture of electron spin: it indicates that the electron spin along the magnetic field is quantized, and can only be in one of two states. Note that "electron spin" is a fundamental measurable quantity, and cannot be visualized (as far as we know) in terms of a "rotating" particle. The theoretical reason for a general particle is essentially that particles cannot have a simultaneously well-defined position and momentum. It is only at macroscopic scales, when effects from the position-momentum uncertainty principle cease to be within our measurement capability, that the interpretation of angular momentum as a rotating rigid body has meaning.
Spin has a subtler importance in quantum mechanics, because it is not predicted from operators that are obtained from classical variables, such as position and momentum. Classically, angular momentum equals r×p, and there is a straightforward quantum generalization, where r and p are replaced by their operator counterparts. However, a deeper interpretation of angular momentum is that it is the "generator" of a system under rotation, just as momentum is the generator of translation. This allows a far wider range of manifestations of angular momentum from a theoretical perspective, because there are many unitary "representations" of SO(3) that cannot be realized through spherical harmonics (the eigenfunctions of the most obvious angular momentum operator). In particular, these extra representations are those with half-integer spin (to be precise, these are actually representations of SU(2), which is a double cover of SO(3): however, observables and transition probabilities associated with SU(2) representations still transform in the proper way under SO(3)).
