chemistry~

Ions themselves cannot conduct electricity. An electrolyte (solution or molten state of ionic compound), consisting of cations and anions conducts electricity. This is the process of electrolysis.
Take molten salt (Sodium Chloride, NaCl (l)) as an example. When you immerse a two connected piece of metals / graphite (the cathode and anode) into an electrolyte and introduce an electric current (a flow of electrons):
Positively charged cation (Na+) will be attracted to the anode which gives out electrons. Na+ ion will get electrons at the anode and changed to elemental Sodium. While the negatively charged anion (Cl-) will gives out electrons at the cathode and change back to Chlorine gas. Hence, the flow of electrons between the two electrodes is maintained, i.e. the electrolyte conducts electricity.
Chemical equations:
At the anode: Na+ (l) + e- --> Na (s)
At the cathode: 2Cl- (l) - 2e- --> Cl2 (g)