In the same way that positive charges move to regions of low potential, negative charges move to regions of high potential. An electric current is a flow of electrons from an area of negative potential to an area of positive potential.
In the animation below, the potential difference is created by a battery.
If we "unwrap" this circuit, we can see that it closely resembles the case of two point charges. If we put a conducting material between two point charges, we will see that a negative charge will build up on the conductor closest to the positive charge. This is because electrons, being negatively charged, flow toward higher potential.
You can see the electrons move to the right, creating a negative charge there, and leaving a positive charge on the other side of the conductor.
In a circuit, the electrons flow through the region of high potential and are "pushed" back down to their original potential when they pass through the battery.
By convention, the direction of "current flow" is opposite the direction of electron flow. For instance, in the circuit above, the electron flow is counter-clockwise, so the current flow would be clockwise.
It is a common misconception that electrons move at nearly the speed of light. Unfortunately, this is incorrect. It is the current that moves at the speed of light, not the electrons themselves.
Now go and answer some questions about opposite charges.