I mentioned yesterday that in a normal wire (copper, 1 mm diameter), a current of 10 amps corresponds to an average electron velocity of about 3.3*10-5 meters per second. To compare, I'd guess that the velocity from thermal energy is around 103 meters per second (within a factor of 10 or so). The only reason the thermal energy doesn't come into play is because it makes the electrons move in random directions, both forwards and backwards. All but the tiniest amount of velocity gets canceled out. The reason the number must be so tiny is that when there are around 1023 of these electrons moving around, a tiny average velocity corresponds to a large overall current.
Even though the electrons move slowly, it is also correct to say that electric current itself travels at the speed of light. If I plug in a lamp, I don't have to wait for an individual electron to slowly crawl through the wire before it lights up. All the electrons in the wire start to move nearly instantaneously. When you plug a wire in, a small electric field appears, pushing the electrons forward. The electrons themselves may not move fast, but the electric field is carried through the entire wire by electromagnetic waves--another word for light. Therefore, electric current travels at the speed of light. (Or close to it anyways. When we talk about the "speed of light" we actually mean the speed of light in a vacuum. Light moves slower when not in a vacuum.)
This post is in response to a question from a reader. Ask and you shall receive. Also, correct my math, because I regularly make mistakes.