In this context, it's the root-mean-square of the electric charge distribution. (Or more precisely, it's related to the slope of the electric form factor at Q^2=0.) There is also at least a magnetic radius (similar to the electric), and a gravitational radius.
Does the electrical charge ever reach 0 or does it keep getting less and less and for practical purposes, we have to set a cut-off limit and count the boundary of the proton from there?
As we understand it, the extent is infinite. However, isolated charges have a hard time staying isolated. They tend to rearrange the charges around them, so that the net effect of their charge becomes effectively zero if you define a vicinity for other particles. I believe the term for this is "shielding."
