In brief, electrons in a conductor move when a potential difference (voltage) is placed on the ends of a circuit, and we describe this movement of charge as electric current.
Explanation:
When one end of a wire (for example) is made negative and the other end positive, electrons in the wire have a force placed on them. They are repelled by the negative end and attracted to the positive end, so they move in the wire, carrying electrical charge. This flow of charge is described as electric current.
The unit of charge is the coulomb (C) and a current of one ampere (I) is flowing when one coulomb of charge passes a point in one second.
Because electrons are negative and it was historically assumed that positive charges carried current, the direction of flow of electric current in a circuit is opposite to the direction the electrons are moving. Electric current is described as moving from the positive to the negative end of the circuit.
Electrons are able to flow in conductors, usually metals, because in metallic bonding the outer electrons of the metal atoms flow freely around between atoms.