potential energy

Potential energy is the energy held by an object (or simply "stored" in an object) because of its position relative to other objects, stresses within itself, its electric charge, or other factors. A more formal way of phrasing it would be the capacity for doing work which arises from position or configuration.

Potential energy essentially tells us how much something wants to move. However having a point with high potential energy is insufficient, you also need a point with lower potential energy, otherwise the object with the potential energy wouldn't move. A book has no use moving sideways on the shelf since all points on the shelf give the same potential energy, but it wants to fall down because at the floor there is less potential energy.

See gravitational potential energy, kinetic energy, elastic energy, electric potential energy.

Why would something have energy due to its position?

An object will "have" energy due to position only if it's in a system with various forces acting to impart force at various locations (a field). A ball at the top of a hill has more potential energy than a ball at the bottom of a hill, specifically because the ball is being acted on by Earth's gravity. If the ball was the same distance away from some other position, but way out in empty space with no forces acting on it, there wouldn't be the same (or probably any) potential energy. Because gravity is pulling down on the ball, it has the "potential" to gain kinetic energy. Also, balls don't just pop into existence at the top of hills; presumably some other agent spent energy to lift the ball up against the force of gravity and put it on top of the hill, and that energy spent should be equal to the potential energy it has now. So, in some sense, the energy isn't "in" the ball at all, it's an emergent property of the system (the ball, the Earth's center of gravity, the hill exerting a normal force so the ball can't fall straight down, etc.)

And it's similar in other kinds of fields. A charged particle in an electric field may have potential energy because it's feeling an EM force which can impart acceleration according to a formula that depends on distance from the source of the field.

When potential energy is constant across the plane, there is no field or force, because if we move an object to any point the work done by the force is exactly zero. The reason why it's zero is that work done in moving an object from one place to the other is exactly the negative change in potential energy. Since potential energy is the same at any two points, there is zero change in potential energy, zero work done, therefore there is no force.