The spin of particle can roughly be physically thought of as the magnetic moment of the particle (or the strength of the magnetic field generated by the particle). For example, a spin-$0$ particle has no magnetic moment, and a spin-$1/2$ particle has half the magnetic moment of a spin-$1$ particle. Experimentally, what this means is that if I shoot a particle while applying a magnetic field, the particle will deflect a certain amount based on its spin. The greater the spin, the more it gets deflected.
Actually, the situation is a little bit more complicated than this due to quantum mechanical effects; there is something called the Lande g-factor which means that a spin-$1$ particle might not necessarily have twice the response under a magnetic field of that of a spin-$1/2$ particle. This, in part, has to do with the fact that the particle (eg the electron) is not actually spinning (even though we call it spin) and there is deep and complicated physics that goes into this.
Regarding the notion of a "spinning particle", it's a bit more accurate to think of spin as the magnetic moment associated with a particle. "Spinning" or orbiting charge does create a magnetic moment, so we call the quantum property spin, but you should keep in mind that this picture is just a cartoon.