Examples include gravity potential, electrostatic potential.

Or, a vector field describing a divergence free vector field (field with only closed field lines) that is its curl: magnetic potential.

Because the physically observable field is a spatial derivative of its potential, adding an arbitrary constant field to it -- a gauge transformation -- will not change anything in the physics of a system. This is called gauge invariance.

In quantum theory, we can identify the potential of a field with the wave function of the intermediary particle associated with that field, like the photon for the electromagnetic field, etc.

In classical mechanics, the force generated by the field is -1 times the gradient of the potential energy (so that the system is pushed towards a lower-energy configuration).