Current is basically a hypothetical movement of positive charge particle from high potential to low potential. Practically electrons moves from lower potential to high.
Now, how can electron move in an atom?
Only in the outermost shell of any atom of metals are free to roam randomly, and once they are subjected to electrostatic field or say a potential difference, they begin to flow in a specific direction, and motion of these elcetron are termed as current.
As protons resides inside nucleous, they are not affected by any external electrostatic field. So atom remains stable, as integrity of atom lies in its nucleous.