What is a potential? Specifically, an electrostatic potential

[MBBphys]

Homework Statement

How do we define an electrostatic potential? My teacher tried to explain it through teaching us gravitational potentials, and I have presented what I came up with under '3.The attempt at a solution'. Please see below and thanks in advance for any help in clarifying whether my understanding seems to be correct, and more importantly, I do not fully understand the words I have underlined and made bold at the end, so if someone could explain it carefully, I would very much appreciate it.

Homework Equations

Please see below.

The Attempt at a Solution

To understand electric potentials, it is useful to first understand gravitational potentials. Not gravitational potential energy (Egpe), but gravitational potentials (Vgrav):

Vgrav = Egpe/m

V1 = mg(h1)/m

V1 = g(h1)

(Egpe = mgh)

Do you see how gravitational potential (Vgrav) is a property of a point in space! There is no mass (m) in the equation Vgrav=gh!

Because of the gravitational field, we can define this property in a point in space known as the gravitational potential, and a mass in that field will gain gravitation potential energy due to the gravitational potential. Different masses will gain different amounts of gravitational potential energy, but the gravitational potential is a constant for any given uniform gravitational field:

Vgrav * m = Egpe
So, in an electrical circuit, the electrons gain electrostatic potential energy, but the electrostatic potential is defined as the electrostatic potential energy per unit charge. The electrostatic potential is a property of a point in the circuit, and a certain amount of charge will have some amount of electrostatic potential energy depending on where in a circuit it is.

 

[MBBphys]

And an additional bit here:
Why does an object gain gravitational potential energy as you lift it up? Because you are doing work against gravitational pull, thus causing the object to increase in the amount of g.p.e it has. You are transferring your bodily chemical energy to gravitational potential energy. Similarly, a battery transforms chemical energy from a reaction to electrostatic potential energy.

Furthermore, just as you are gaining g.p.e. because you are doing work against gravitational pull, the electrons gain electrostatic potential energy because they are being repelled by the positive terminal and are attracted to the negative terminal.