At the cathode

Rules 3 and 4 say what happens once the positive ions reach the negative electrode.

Rule 3

Positive ions travelling to the negative electrode gain electrons and are reduced. If there are several kinds of positive ion in the solution, the one usually more likely to be reduced is the cation of the element lower in the activity series.

Two examples

If a solution has Na⁺ and H⁺ ions, the H⁺ ions — formed by hydrogen, which is below sodium in the activity series — gain electrons and are reduced.
If a solution has Cu²⁺ and H⁺ ions, the one that gains electrons is Cu²⁺ — formed by copper, which is below hydrogen in the activity series.

2H⁺ (aq) + 2e → H₂ (g)

Cu²⁺ (aq) + 2e → Cu (s)

the two cathode reactions of the examples

Rule 4

Because a reduction half reaction happens at the negative electrode, the negative electrode is the cathode.

Why the lower metal wins

A metal high in the activity series, like sodium, holds on to its electrons loosely — it gives them away easily. Turn that round: its ion, Na⁺, has very little pull on electrons and is hard to turn back into the metal. An element lower in the series, like hydrogen or copper, keeps its electrons more firmly, so its ion takes electrons back readily. At the cathode, the ion that takes electrons most readily is reduced first. This is why electrolysing salt water gives hydrogen, not sodium.

The cathode's sign has flipped

In an electrochemical cell the cathode is the positive terminal; in an electrolytic cell it is the negative electrode. What never changes is the reaction: reduction is always at the cathode.

Reduction at the negative cathode

In electrolysis the negative electrode is the cathode. Of several cations, the one from the element lower in the activity series is reduced.