A practical model for a metal is a three dimensional grid of positively charged nuclei surrounded by a cloud of electrons that are free to move about the object. The electrons in the cloud respond quickly and easily to an applied electric field.
When you place a difference of electric potential (a voltage) across a piece of metal, an electric field is created that causes the electrons to move giving rise to relatively large current.
In contast, a nonconductor or insulator has few or no free electrons and so will have little or no current flow. That is until you raise the voltage so high that it creates an electric field so strong in can begin to rip bound electrons from the atoms. The insulator is then said to break down.
For example, air is poor conductor until the electic field reaches about 300,000,000 v/m. Then sparks fly.