When the free electrons move from atom to atom of a conductor or semiconductor, they typically bounce around randomly. Once a voltage is applied to the material, the electrons move from the negative towards the positive.
Remember that electrons have a negative charge. Since the negative side of the voltage source repels the electrons, they move away from it and head towards the positive side, which they are attracted to.
Current, in electrical terms, is the rate of the flow of charge. The current in a conductor is measured by the amount of charge (number of electrons) that flows past a certain point in a unit of time:
I =Q / t
An ampere, or amp, is the unit for current. This is symbolized by A.
The short video below does an excellent job of explaining current and voltage.