Every metal has a large number of free electrons which are in a state of continuous rapid zig-zag motion. But their directions of motion are so randomly distributed that the average velocity of electrons is zero. So, there will be no net flow of electrons in one direction.
When some potential difference is applied across the ends of a conductor, an electric field is set up inside the conductor and free electrons are accelerated from negative end to the positive end of the conductor. On moving, they suffer frequent collisions against the ions, and lose energy. During the short time between every two successive collisions, the electron accelerates towards the positive end and describes a curved path between two successive collisions. But the extra velocity so gained is destroyed at each subsequent collision. The net result is that in addition to their random motion, the electrons acquire a small velocity towards the positive end. This average velocity is called drift velocity.
Thus, drift velocity is defined as the average velocity with which free electrons get drifted towards the positive end of the conductor under the influence of an external electric field.