Electrolysis

A galvanic cell runs on its own because the reaction is spontaneous: electrons flow downhill, releasing energy as current. Electrolysis is the reverse setup. You attach a battery or power supply and force electrons to flow the wrong way, driving a reaction that would never happen by itself. The battery does work on the system, so free energy increases instead of decreases. The amount of product you get depends on how much charge passes through, measured in current times time, then converted to moles of electrons.

When several species could be reduced or oxidized, the one that requires the least energy wins. At the cathode, that means the species with the easiest reduction potential gets reduced first. At the anode, the species that is easiest to oxidize loses electrons first. In molten NaCl, there is no water at all, so the only players are Na+ and Cl-. Na+ is reduced to liquid sodium metal at the cathode, and Cl- is oxidized to Cl2 gas at the anode. That is the clean, textbook reaction you expect from the formula.

Aqueous NaCl changes the cast of characters. Water is far easier to reduce than Na+, so at the cathode water gets reduced instead, producing H2 gas and leaving OH- behind, while sodium ions stay dissolved as spectators. At the anode, Cl- is still oxidized to Cl2, because it remains easier to oxidize than water under these conditions. This is why salt water electrolysis bubbles H2 and Cl2, never sodium metal, and it sets up electroplating, where you choose electrode and solution so the metal you want is the one that deposits.