Surface charge and surface discharge in batteries refers to the superficial charging and discharging which occurs only on the surface of the electrodes. A lead acid battery consists of lead oxide anode and a lead cathode. When discharging, the lead oxide in the anode and the lead in the cathode get converted to lead sulphate. When charging, the reverse happens.
Surface charge refers to a condition when the chemical changes mentioned above occur only in the surface of the electrode. For instance, if only the lead sulphate in the surface of the anode of a discharged battery gets converted into lead oxide, the battery will indicate a full charge when the open circuit voltage is measured. However, the charge in the battery will last only for a short time.
Surface discharge is a condition when the open circuit voltage of the battery wrongly indicates a discharged condition, when the battery is still holding charge. This too relates to chemical changes which occur superficially on the surface of the electrode.
The surface charge can be removed by applying a slight load on the battery. In vehicles, switching on the headlights for a while can help remove the surface charge.