Electric potential
The electrostatic potential energy per unit charge at a point. A scalar field measured in volts. V = −∫E·dℓ from a reference point.
Definition
Electric potential V is the electrostatic potential energy per unit charge. If you would have to do work W to bring a small positive test charge q from infinity to a point P, then V(P) = W/q. Units: joules per coulomb, called volts. Because work is a scalar (just a number), potential is a scalar field — one real number at every point in space — which is enormously easier to compute with than the vector electric field.
Once you have the potential, you can get the field by differentiation: E = −∇V, the negative gradient. The minus sign means the field points from high potential toward low potential, the same way water flows from high ground to low. Moving a charge around changes its potential energy by qΔV; two points with no potential difference between them do no work on a charge that moves between them, regardless of route.
Only potential differences matter physically. The absolute zero of potential is a convention: in electrostatics problems it is usually set at infinity; in circuits it is usually set at "ground" — a particular chunk of copper arbitrarily declared to be 0 V. This is why the output of a battery is specified as a voltage (a difference) rather than a single potential, and why connecting one terminal of a battery to "ground" and nothing else does nothing — voltage only does work when it drives current through something.