In a conductor, valence band and conduction band overlap each other as shown in figure. Therefore, there is no forbidden gap in a conductor.
A small amount of applied external energy provides enough energy for the valence band electrons to move in to conduction band. Therefore, more number of valence band electrons can easily moves in to the conduction band.
When valence band electrons moves to conduction band they becomes free electrons. The electrons present in the conduction band are not attached to the nucleus of a atom.
In conductors, large number of electrons are present in conduction band at room temperature, I.e, conduction band is almost full with electrons. Where as valence band is partially occupied with electrons. The electrons present in the conduction band moves freely by carrying the electric current from one point to other.