Gear ratio explained. The gear ratio, is the ratio of turns made by one gear, compared to the other gear, when the gears are connected by a chain. An example of gear ratio will be with one gear of 10 teeth and the second gear of 20 teeth, giving a ratio of 2 (20/10=2). To calculate the ratio, devide the number of teeth on the rear sprocket, by the number of teeth on the front sprocket and the resulting number is your gear ratio.
Why do you need to know your gear ratio? It might be, that your bike is not performing as you would like and a change in gear ratio might help the performace.

GEAR DOWN. An example of this would be, that the bike is difficult to start in first gear and the clutch needs to be used more than you would like. You have plenty of top end speed, but need more slow speed control. The answer, might be to GEAR DOWN the bike. You can do this by adding more teeth to the rear sprocket or reducing the number of teeth on the front sprocket.

GEAR UP. The opposite situation, might be that the bike has lots of power, but is reaching the rev limiter in top gear and you would like to reduce the revolutions at a particular speed. To do this, you would need to GEAR UP the bike. By reducing the number of teeth on the rear sprocket or increase the number of teeth on the front sprocket. Diameter of gear. Each tooth added or taken away from a sprocket, will reduce the diameter of the sprocket by a given amount. For simplicity, you can think about 5mm change on the diameter, for each additional or reduced tooth. This will help understanding in relation to space requirements for gear ratio changes. Chain length. The addition of teeth to the sprocket, may require a longer chain or changing the position of the rear wheel, by moving that backwards or forwards. It will depend on the addition or subtraction of sprocket teeth. This can be calculated by considering the chain pitch. For simplicity, you can think about each chain link requiring 16mm length (actually 520, 525, 530 chains require 15.875mm per link to be exact). But the chain is going around the rear sprocket and the front sprocket, so the important number would be the change in diameter of the sprocket and not the chain length. An example would be to remove one tooth from the rear sprocket. This would make the chain more loose and to correctly adjust the chain, you would need to move the rear wheel backwards (away from the small sprocket) by 5mm. This movement would compensate for the change in diameter. Due to chain construction, it is not possible to remove only one link and it is neccasary to remove even numbers of links (2, 4, 6, 8 etc).