It means that under the conditions of the measurement the zeta potential of the particle is positive or negative and thus would attract to a surface or particle of the opposite sign.
The sign of the Zeta Potential (+ or -) is the sign of the overall charge of the particle. As such it is due to the combined effect of the surface charge plus any specific ions that may be loosely adsorbed at some distance from the surface but nevertheless travel with the particle. The magnitude of the Zeta Potential gives an indication of the stability of a colloidal suspension because high values (+ or -) result in strong repulsive electrostatic forces among the colloidal particles. The environmental variable that has the greatest effect on both the sign and magnitude of the Zeta Potential is pH, with most environmental particles acquiring a positive surface charge at low pH and a negative surface charge at high pH.
I am not an electro-chemist and therefore you will get a physicist's view on the matter :
1. Consult R.Feynman Lectures, Vol.II, p.7-9. The essential physics is explained there, inclusive the characteristic Debye length ( exponential decay of the potential away from the charged colloidal particle).
2. Zeta potential IS NOT a physical quantity that can be measured directly and therefore it depends crucially on the model you use to describe it, if it exists at all.
3. The apparent consensus within electrochemistry is that it is a potential at a distance (slip plane) from the surface of the colloidal particle that separates particles(ions) that have stuck to the colloidal particle from those that are free to move. In other words the colloidal particle effective size somewhat increases and the slip plane is a measure of this. In aqueous pure ion solutions (like NaCl for example) this is called hydration shell . If the colloidal particle is roughly spherical, this effective radius defines the electrical mobility of the colloidal particle (only if the total charge of particle and the stuck ions is non-zero !!) and can be measure through Electrical Impedance Spectroscopy (EIS), using Einstein-Stokes relationship .
4. The other experimental methods are just approximations to EIS method.
5. As for the sign of zeta potential, I agree with Francois, but be aware that the physics/electrochemistry of the colloidal particle-solution interface is quite complicated as can be deduced from , as Francois points out, the pH factor (concentration of the H+ ions in the solution) dependence of zeta potential.