The state of a particle will generally change if you rotate it. The details of how the state changes under an infinitesimal rotation are contained in the angular momentum operator J. This operator can be divided into two parts: an "orbital" angular momentum and a "spin" angular momentum.
The orbital angular momentum tells you how the state changes under a rotation due to wavefunction of the particle. I think this is pretty intuitive: if the wavefunction has some angular variation, then rotating the particle will change its state, so there ought to be some contribution to J due to the wavefunction. Spin angular momentum on the other hand is a lot stranger, at least to me. If you rotate a particle with spin, its state will change even if its wavefunction is completely isotropic.
Maybe I'm being naive, but to me this fact implies that particles cannot be point-like but must have some extended structure. If particles were really point-like, their only degrees of freedom would be their position, so how could something like spin arise?
Could string theory explain spin then? Or is spin introduced into string theory in as ad hoc a way as it's introduced into non-string physics?