Flow of electrons in a wire
In a metal like copper some of the electrons are not bound to individual atoms. If a voltage is applied across a copper wire these free electrons flow from one atom to the next. This flow of electrons is an electric current but the copper atoms themselves do not move so the copper wire does not vanish. Of course, the voltage source must supply additional electrons for the flow to continue.

As an analogy, think of a waterfall -- the electric current is like the water molecules falling, and the water molecules are falling due to gravity (which is analogous to voltage). The waterfall must be supplied with more water molecules for the waterfall to continue to exist, but the riverbed over which the water molecules flow (analogous to the copper atoms) do not move or vanish.