It means that the spins of a collection of protons are correlated. In the ideal case (100% polarization) they would be all in a single direction. In the case of a proton beam, the polarization is generally along the direction of flight (positive helicity) or against the direction of flight (negative helicity).
In most matter the spins are randomly aligned so the material is said to be "unpolarized".
For the most part it does not make sense to talk about the polarization of a single particle, but if you are going to do it anyway, you must have preferred axis against which to measure it.
I guess the spin of the proton is obtained from the vector addition of the 3 quarks' individual spins, so it can't be 0
That is a reasonable guess, but most of the proton's angular momentum comes from sources other than the spin of the valance quarks.