If electrons move slowly in an electrical circuit then what signal or energy is it that travels at the speed of light?
It is changes in the fields inside and around the wire that travel at the speed of light. Imagine the wire as a hollow tube full of electrons. When there is no current the electrons are all sitting there, repelling each other, but since there is nowhere to go, they just sit still.
When an electron at the back of the wire starts getting pushed (by a battery for example), it gets closer to its neighbors in front, which then pushes them forward. These electrons start to move, which then causes them to start pushing on their neighbors. Eventually all the electrons are moving down the wire.
The speed with which the electrons hear about their neighbors moving determines how quickly the signal travels down the wire. Nothing about this process actually requires the electrons to actually go anywhere quickly, just that their neighbors feel changes quickly.
This is all a huge simplification, in reality there is a continuous electric field, and there is a magnetic field generated around the wire, and of course the wire isn't hollow, but this picture can help for getting the concept.
Could we analogise it to something like this? Imagine a really long train with a huge number of coaches. The engine begins to move, very, very slowly. Instantly, the last coach also begins to move. The speed of the engine has no relation whatsoever to the speed at which the information of the train starting to move is propagated.
The electrons are like the coaches.
Electrons themselves move at the glorious speed of fractions of a millimeter per second. However, they are so close to each other that they are constantly bumping in to each other. This makes an electrical signal travel down a wire at somewhere usually around 2/3rds the speed of light. That speed can be effectively slowed by various circuit elements as well. A signal inside a coaxial cable will travel more slowly than a signal in a free-hanging wire. See wikipedia on the Velocity Factor for more details about this, specifically part about the velocity factor in a lossless transmission line.
As for "electrical effect", he's probably talking about the electromagnetic radiation that can be produced by certain oscillations in circuits (changes in the electrical field resulting in radio waves, electrons moving between energy states in LEDs, etc). These travel at the speed of light in a vacuum because electromagnetic radiation is light and vice versa.