To elaborate on Emilio's nice answer, let me add that spin is, as he says "an internal degree of freedom." What does this mean? One way to think about it is that its an intrinsic property of the particle in an analogous way to how mass is an intrinsic property of massive particles or charge is an intrinsic property of charged particles.
You might then ask "well mass tells us how difficult it is to change the state of motion of particles, and charge tells us how strongly a charged particle interacts with electromagnetic fields, so what does spin tell us?" As Emilio writes, the spin is related to how much of a particular kind of angular momentum a particle can have, not necessarily due to the fact that it is actually spinning.
In light of this, I personally feel that it's dangerous to identify spin with "which way the particle is spinning" since it's unclear, in the case of certain elementary particles like electrons, whether the particle is "spinning" in some sense, or what that would even mean if the particle is modeled as being truly pointlike.