Definition/Summary
Potential energy is simply another name for (minus) the work done by a conservative force.
Since the work-energy theorem states that change in energy minus work done is constant, that means that for a conservative force, energy plus potential energy is constant.
For example, a object of mass $m$ moving a height $h$ and distance $s$ along a curved path up a slope loses energy $\int mg\,dh$ because of the force of gravity and $\int \mathbf{F}\cdot d\mathbf{s}$ because of the force $F$ of friction. Gravity is conservative (while friction is not) and so the first integral may be replaced simply by $mgh$
Potential energy is measured relative to an arbitrary level, which in each case may be taken to be whatever is most convenient (for example the starting level, or infinity).
Potential energy (commonly shortened to PE) is a scalar (an ordinary number), with the same dimensions as energy...