Understanding the concept of the gear ratio is easy if you understand the concept of the circumference of a circle. Keep in mind that the circumference of a circle is equal to the diameter of the circle multiplied by Pi (Pi is equal to 3.14159...). Therefore, if you have a circle or a gear with a diameter of one inch, the circumference of that circle will be 3.14159 inches. The following figure shows how the circumference of a circle with a diameter of 1.27 inches is equal to a linear distance of 4 inches:

Let's say that you had another circle whose diameter was 1.27 inches / 2 = 0.635 inches, and you rolled it in the same way as in this figure. You would find that, because its diameter is one half of the circle's in the figure, it has to complete two full rotations to cover the same 4 inch line. This explains why two gears, one half as big as the other, have a gear ratio of 2:1. The smaller gear has to spin twice to cover the same distance covered when the larger gear spins once.

Most gears that you see in real life have teeth. The teeth have three advantages:

1. They prevent slippage between the gears - therefore axles connected by gears are always synchronized exactly with one another.
2. They make it possible to determine exact gear ratios - you just count the number of teeth in the two gears and divide. So if one gear has 60 teeth and another has 20, the gear ratio when these two gears are connected together is 3:1.
3. They make it so that slight imperfections in the actual diameter and circumference of two gears don't matter. The gear ratio is controlled by the number of teeth even if the diameters are a bit off.