Traveling electric charge is called an electric current, or just current. There is only a current through a conductor if there is driving force directing charge in a particular direction. This driving force is typically called a voltage or electrical potential. The conductor also must form a closed circuit, otherwise charge would build up at one end and produce a potential pointing in the opposite direction and canceling out the first potential.

In a conductor that is not part of a circuit, there is no current. One could argue that the electrons are still moving, but their motion is random and the net result is no real "traveling" charge.