|What happens to the current when it "stops"?
Current refers to moving charged particles. In most solids, the particles that do the moving are negatively charged electrons that move in the opposite direction from the way we say that current is flowing. These charged particles are the components of atoms and molecules, so they are always there inside a wire or the filament of a light bulb, even if they are not moving. Thus when the current "stops", these electrically charged particles simply stop moving. You can imagine a pipe full of water. The water can be flowing to the right or left (a current) or it can be standing still (no current). The water itself, like the charged particles, doesn't disappear when the flow stops.