When there is no potential difference (ie the electrostatic condition) there is no electric field inside the conductors and the electrons or charges will be moving in random direction because of their own thermal energy. This thermal energy depends upon the temperature of the conductor. But still the average velocities of electrons is zero because there is no particular direction of motion and all the charges cancel each other's velocity.
When you apply a potential difference across the conductor (ie the electrodynamic condition) still there is no electric electric field inside the conductor but there is an potential difference across the conductor which produces an external electric field due to which -ve charges start flowing in direction of electric field and +ve charges in opposite direction .The net charge inside is zero but now the average velocity is not zero because now all electrons are flowing in direction of electric field only. So due to flow of these electrons a current is developed in the conductor.