In aqueous medium the oppositely charge ions present in solution are charge carrier for conduction. They move towards oppositely charged electrode and get discharged there making chemical change of the substance dissolved and thus conduction occurs .
Some polar covalent molecules when dissolved in polar solvent, water may get ionized in the medium. As in case of Hydrogen Chloride gas , it gets ionized in aqueous medium as
This solution becomes conductor of electricity due to the presence of ions in it. But pure Hydrogen Chloride in liquid state or in dissolved state in non-polar solvent like