Electric Potential and Potential Difference

Electric potential at a point in an electrostatic field is defined as the work done by an EXTERNAL agent, per unit positive test charge, in bringing that test charge from infinity (where the field, and by convention the potential, are taken to be zero) to the point in question, WITHOUT any change in the charge's kinetic energy (i.e. moved slowly, quasi-statically, so that at every instant the external force exactly balances the electrostatic force on it):

Electric potential is a scalar quantity -- it has a magnitude and a sign, but no direction -- and its SI unit is the volt ($\text{V}$), defined so that $1\ \text{V} = 1\ \text{J/C}$: a potential of one volt at a point means one joule of work is needed to bring one coulomb of positive charge there from infinity.

Potential difference between two points $A$ and $B$ is defined similarly, as the work done per unit charge in moving a test charge from $A$ to $B$:

This is, in practice, the more directly useful quantity, since only DIFFERENCES in potential (never some single "absolute" potential) can actually be measured by an instrument such as a voltmeter -- the choice of taking $V=0$ at infinity is simply a convenient, universally agreed reference, exactly analogous to choosing sea level as the zero of height in a gravitational problem.

A crucial fact, without which the very definition of potential above would not even make sense, is that the work done in moving a charge between any two points in an electrostatic field is independent of the path taken -- a direct consequence of the electrostatic force being a conservative force. If the work done depended on the path chosen, "the potential at point $P$" would not be a single well-defined number at all, since different routes from infinity to $P$ could give different answers; path-independence is exactly what guarantees $V(P)$ is one definite value for every point $P$. …