Electricity happens when electrons start moving around. Running or spinning a magnet past a copper coil makes them start moving around in the copper coil. So the axel of the spinning part is attatched to a wheel on the inside which has either the magnets or the copper coils on it. The other component stands still.

The electricity created by the copper coil will flow through wires attatched to either end. You can charge a battery (do not ever try charging a battery unless you have an extensive knowledge of electronics or a kit or something because if it is overcharged it could explode), run a motor, turn on a light, etc. A gigantic wind turbine or a hydro dam (which also uses a turbine) can hook up to the grid and power a whole neighborhood!