That's how most generators work. A magnet spinning in a coil of copper wire, or vice-versa, a coil of copper wire spinning in a magnet. Most of the time both are coils of wire and the magnetism comes from putting an electrical current through one. If a magnet moves past a wire or a wire moves past a magnet it induces a current in the wire. Make that wire a complete circuit and you have electron flow. The amount of current is dependent on several things, but the more windings of wire that are passing through the magnetic field the more current you get. In the US our power's frequency is mainly 60 Hertz, which is related to the speed of the rotor. It's not always like that, some generators regulate the frequency after that so it stays constant, they would be the ones you should use with frequency and voltage sensitive tools like computers. Some are a little more complicated than that to make clean power.
I see a heck of a lot of generators at work, we sell them and I've repaired them for years. This was a little simplified and they are manufactured several different ways but I just wanted to give people a little info for generator buying. After that it's all about matching up your wattage.
The more you know!
P.S for gotta do it nerds: You can check it out on a small scale if you have a decent multimeter, a magnet, and a piece of wire. Moving the magnet back-and-forth across the wire will induce a small current in the wire which can be read with the meter's amperage setting if it can read a very low amperage. You could even put the leads of the meter together and use that as your wire. You made power!