Drift velocity is the velocity at which each individual electron travels inside a wire carrying current. You'll find that this velocity is actually VERY tiny, usually a few millimetres per second.
So then the question is why does an electric signal travel so fast in a wire (roughly a third of the speed of light, in a typical copper conductor), if the drift velocity is very small. The reason is that in a wire carrying a current, one electron pushes another electron, which in turn pushes another and so on, it is not just the same electron travelling from one end of the wire to the other.
A useful analogy to think of it is like dominos. If you set up some dominos stacked behind eachother and you give the one at the front a small push, then it will push the one in the front of it, which will then push the one in front of it and so on. Even though the speed at which each individual domino is tiny (only a few millimetres a second, like drift velocity), all of the dominos stacked together fall down very fast. If you line up the dominos in a straight line a few metres long, it will only take a couple of seconds until the last domino falls down despite the fact that each individual domino moves at a few millimetres per second.
The formula for working out drift velocity is:
v = I/nAe
v = drift velocity (m/s)
I = current (A)
n = no of conduction electrons per volume (electrons/m^3)
e = charge of the electron (C)
A = cross section area of the wire (m^2)
