How does electricity actually flow through wires?

The answer

Individual electrons in a wire drift incredibly slowly - only about 0.1 millimeters per second. What travels near the speed of light is the electromagnetic wave that pushes electrons along, like a chain reaction of billiard balls. When you flip a switch, the electric field propagates through the wire almost instantly, causing electrons everywhere in the circuit to start moving at once. The energy is carried by the electromagnetic field surrounding the wire, not by the electrons themselves.