Normal electrical wires conduct electricity by pushing free electrons through solid metal. Electrolytic conduction works completely differently because it relies on the movement of whole atoms. When you dissolve a salt in water, it breaks apart into positive and negative ions.
Applying a voltage across this liquid creates an electrical field. The positive ions, called cations, swim directly toward the negative electrode. The negative ions, called anions, travel in the opposite direction toward the positive electrode.
This physical movement of charged particles creates a complete electrical circuit through the liquid. Electrons never actually swim through the water between the two electrodes. The electrons only travel through the metal wires on the outside of the container.
The total conductivity depends heavily on the concentration of dissolved ions in the liquid. It also depends on the charge of the ions and how easily they can move. Interestingly, heating an electrolytic solution actually increases its ability to conduct electricity.
A higher temperature makes the liquid thinner and less sticky. This lower viscosity allows the charged ions to swim through the fluid much faster. This simple principle allows scientists to build working batteries and fuel cells.