Qualitatively, think of the overall motion of electrons as having two components.
One component involves the mobility of free electrons. That is the very rapid random motion of the individual electrons in a conductor due to thermal energy, for which there is no net collective motion of the electrons in any particular direction. This random motion is related to the mobility, μ, in Krishna's answer.
The second component occurs when an electric field is introduced. The electrons now collectively move toward the direction of the "positive end of the conductor" as you put it. Or, more formally, the electrons collectively move in a direction opposite to the direction of the electric field in the conductor. By convention, the direction of the electric field is the direction of the force (and movement) that a positive charge would experience if placed in the field.
The collective motion of the electrons is called the drift velocity and is directly related to the current in the conductor. This velocity is very slow compared to the random velocities of the electrons that occur with or without the electric field present.
Doesn't this contradict the the motion of free electrons in an
conventional current which is due to applied electric field in
No it doesn't. Because drift velocity, the motion of the electrons collectively which is very slow, is independent of the mobility, which is the random velocities of the individual electrons. Let me give you rough analogy.
Suppose you are on the shore and able to see a school of fish where the individual fishes are freely and rapidly darting about in no particular direction. That's roughly analogous to the mobility. But then you also see the whole school moving slowly from left to right because of the ocean current moving in that direction. That would be their collective drift velocity.
Hope this helps.