The valence electrons in metal atom are free to move. So the metal contains large number of free electron which are moving randomly in all direction from atom to atom. When an electric field is not applied to a metallic conductor, the free electrons are in thermal equilibrium with the conductor and are in random motion on the surface . So, the average velocity of the electrons in a direction is zero. Since this motion does not make up transport of a net charge across any section of the conductor, there is no current in the conductor.
When an electric field is applied to the conductor by connecting a battery across it , each electron is acted by an electrostatic force, and the electrons get accelerated in the opposite to that of the field. Then the electrons gain velocity and kinetic energy. These electrons however collide with atoms (or ions) on lattice site of the metal. During the collisions, the electrons give up their energy to the atoms and so their velocity decrease. However, the electrons again accelerate due to the electric force and make collisions with atoms. As a result of the repeated collisions, the average acceleration of electrons is reduced to zero and the electrons thus acquire a constant average velocity opposite to the direction of the electric field. This velocity is called the drift velocity which is responsible for the flow of a current through the conductor.
Random motion of an electron in a metallic crystal in the absence of electric field.
Thus the average velocity acquired by the free electrons in a conductor subjected to an electric field is called drift velocity.