"Spin" doesn't really mean something is spinning. Electrons as far as we can tell are point-particles, so it doesn't make sense to say they are spinning.
But particles do have some intrinsic angular momentum associated with them, and we refer to that as spin. It's just there. You can't stop it like you could stop a ball from spinning.
Electrons do "move" around atoms and gain extra angular momentum (like Earth orbiting the Sun), but in that case it depends on which electron shell the atom is in and in what position. And if you take the electron away from the atom, the angular momentum is gone. You can't do that with Spin.
Moreover, all particles have spin. Quarks, Electrons, Protons, and Neutrons have Spin 1/2. Photons and Gluons have Spin 1. There are some neat rules regarding Spin. If two particles have Spin 1/2, they can't be in the same state (i.e. energy level and position). That's why you can only have 2 atoms in the first electron shell of an atom. One electron is Spin +1/2, the other Spin -1/2. When you move up in electron shells, they start going to different places. This is just a flavor, though, and I'm really simplifying it.
But particles do interact with each other based on what their Spin is, because that's where magnetism comes from. So it makes sense that a single particle could determine something, because whether its Spin is +1/2 or -1/2 means it would affect surrounding particles differently.
Of course, it's still just one tiny particle...