thermodynamics - how is the relation between Gibbs free energy and cell potential?

First of all what is exactly Gibbs free energy in regards to an electrochemical process? How can we visualise it? How does Gibbs free energy change in an electrochemical reaction? How is it related to the cell potential?

Isn't potential the amount of work done per unit charge to bring it from infinite to that point?

Now the ionisation in cell can be thought of as occurring in multiple steps, for example:

$$\begin{aligned} \ce{Zn(s)} & \ce{->Zn(g)} && (1) \\ \ce{Zn(g)} & \ce{->Zn+(g) + e-} &&(2) \end{aligned}$$

Then the work done to remove the charge should be the change in Gibbs free energy of second reaction and that energy should be equal to potential energy, but we take the combined change in Gibbs free energy as change in electric potential energy of the electron. Why is that so?