What's the Relation between Potential of mechanics and electricity?

The electrostatic potential is a potential just like any other mechanical potential. Since the force on a particle with charge $q$ is $vec F_text{stat} = qvec E$, the potential $phi'$ that gives the force as $vec F_text{stat} = - vec nabla phi'$ is just the electrostatic potential (the "voltage") times $-q$. The electrostatic force is a perfectly ordinary conservative force.

If you have currents, however, the full electromagentic force is not conservative in the classical mechanical sense. It has a generalized kind of potential, though, which is the four-potential of electrodynamics.