A points system used on the Xbox 360. Often abbreviated as just 'G'. Every Xbox 360 game has "achievements" and when you unlock these achievements you will get a certain amount of gamerscore.
For example, one achievement on Gears of War is called "Commando" and you unlock this when you complete the game on the hardest difficulty. It is worth 30G.
Your Gamerscore is shown on your Gamercard