The drift velocity is the average uniform velocity acquired by free electrons inside a metal by the application of an electric field which is responsible for current through it.
In absence of any electric field [Fig.(A)], the random motion of electrons does not contribute to any current. The number of electrons crossing any plane from left to right is equal to the number of electrons crossing from right to left (otherwise metal will not remain equipotential). When an electric field is applied, due to electric force the path of electrons in general becomes curved instead of straight lines and electrons drift opposite to the field [Fig.(B)].
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