Electrolysis - What is the basis for the discharge rules?
The redox potential is a measure of tendency of  redox system to exist rather in oxidized form ( negative potential values) or reduced form ( positive values ).

Therefore the higher the standard redox potential is, the stronger oxidation effect and weaker reduction effect have the respective oxidized/ reduced forms of the redox system.

And vice versa.

Particularly, on the opposite sites of the standard redox potential table are
$$\begin{align}\ce{
Li &amp;&lt;=&gt;&gt; Li+ + e- \\
F2 + 2e- &amp;&lt;=&gt;&gt; 2 F-  \\}
\end{align}$$

The former is very strong reducting agent, while latter is very strong oxidating agent.

I will illustrate it on anode case, with the cathode case it is the same, but with the reversed potential order.

When the anode potential is externally forced to raise up, if first start to oxidate the reducted form of the redox system with lowest redox potential.

If the current is already saturated by diffusion rate of redox reagents, futher increase of potential does not increase the current until it reaches potential region of next easiest to oxidize redox system.

If the redox systems have similar values, the next system start to react on the electrode before the system 

The result is, the curve current versus anode potential has a typical look of the  rounded stepwise  voltammetry curve.

See also Voltammetry,  Polarography and example of the cathodic polarographic curve.