I think it's kind of just an abstraction to make it easier to think about and solve problems. Really, the basic thing here is the force between two charges, given by Coulomb's Law. From that, we can begin to talk about an electric field (force per unit charge)---something that can exist with only one charge (whereas Coulomb's Law requires at least two charges to find a force). That electric field tells us something about how other charges will move when placed in the electric field (similar to the concept of a gravitational field, which tells us how matter will move). But, electric fields can be messy, as they are vectors. As you learned in your mechanics class, sometimes it is much easier to solve a problem if you forget about forces, and just make use of conservation laws, one of the most commonly used being the conservation of energy. The concept of energy allows us to solve some problems using just scalar quantities. We can consult Coulomb's Law again to calculate an electric potential energy for two charges, using the concept of work (integral of force dotted with displacement). Looking at changes in energy between multiple charge configurations can be helpful for problem solving, but what if we would like a more general description of what would happen if we add a charge to the system? Then it would be nice if we didn't have to always do our calculations with at least two charges. One would suffice. Or we could have some kind of a charge distribution. Both of these, of course, would create an electric field. But if we didn't care about vectors to solve our problem, we could compute an electric potential energy per unit charge. Then we can predict the behavior of any charge that we add to the system. (The charges we add will try to move from areas of high potential energy to low potential energy, or, equivalently, from areas of high electric potential to low electric potential (for positive "test" charges) and from areas of low electric potential to high electric potential (for negative "test" charges).)
I don't know that an electric potential is a "real" thing. It is a useful, abstract quantity derived from the real thing, which is Coulomb's Law. Perhaps the best way to think about an electric potential is as a sort of elevation. Moving to lower potential is like moving down a hill (for positive charges). Moving to higher potential is like climbing a hill---positive charges will require either some initial kinetic energy or some work done on them to move up these "hills".
Well, hopefully that is a bit helpful.
