Actually zmmmmm is correct about the speeds. The drift velocity of electrons in a conductor is quite low. Somewhere around 1 meter per second (its been a long time since my physics degree so it may not be exactly 1ms-1).

But what happens when you close a circuit of a conductor across a potential difference is an electric field is set up surrounding the conductor. This electric field accelerates all the electrons in the conductor at once. This is why even though the drift velocity of an electron in a conductor is very low, when you switch on a light switch the bulb lights immediately.

The propagation of the electric field is at the speed of light (its electromagnetism). So "electrons" move slowly but "electricity" moves at the speed of light.

Update: found a wikipedia page on it: http://en.wikipedia.org/wiki/Speed_of_electricity