an aqueous solution of acid shows acidic character.
In presence of water, it dissociates into Hydrogen ions or protons. The presence of these free ions helps it to conduct electricity.
When electricity is passed through the acid (that already has H+ ions or protons present in the solution) the protons, being positively charged, reach the cathode where they pick electrons and get converted into hydrogen gas. This forms a complete electrolyte reaction and conduction of electricity, hence, is possible.