The Arduino Leonardo uses the ATmega32U4, which has circuitry on board that handles USB (device) functionality such as mouse and/or keyboard (when you bypass the Arduino library all sorts of USB devices are possible, such as joysticks, midi, virtual serial, audio in/out etcetera).
The ATmega328 on the 'standard' Arduino and Gertboard unfortunately doesn't have that circuitry in the chip.
There are software solutions available though. Most used is the obdev USB stack. It works surprisingly well: http://obdev.at/products/vusb/index.html
I'm not sure how easy it is to compile it with the Arduino IDE. You may have to resort to a good old makefile to compile and upload it (let us know if you run into trouble with that, should be an easy fix).