The engine core (hot part) works best at a different speed than the fan part - fast and hot, vs slow and cool. hence they have a gearbox. But it's not like a cars gearbox, it has a single gear.
What the current generation of engines have is two or three co-axial shafts rotating at different speeds.
The difference in shaft speeds is achieved by varying the size, shape and placement of the turbine blades connected to the shaft, not with a gearbox.
(And a single gear in isolation would be pointless. By definition, the purpose of a gear is to mesh with another gear.)
Here's an informative video: http://www.youtube.com/watch?v=zy4A-z2WKhw