If you want to create a high gear ratio, nothing beats the worm gear. In a worm gear, a threaded shaft engages the teeth on a gear. Each time the shaft spins one revolution, the gear moves one tooth forward. If the gear has 40 teeth, you have a 40:1 gear ratio in a very small package. Here's one example from a windshield wiper.
A mechanical odometer is another place that uses a lot of worm gears:
There are many other ways to use gears. One specialized gear train is called a planetary gear train. Planetary gears solve the following problem. Let's say you want a gear ratio of 6:1 with the input turning in the same direction as the output. One way to create that ratio is with the following three-gear train:
In this train, the blue gear has six times the diameter of the yellow gear (giving a 6:1 ratio). The size of the red gear is not important because it is just there to reverse the direction of rotation so that the blue and yellow gears turn the same way. However, imagine that you want the axis of the output gear to be the same as that of the input gear. A common place where this same-axis capability is needed is in an electric screwdriver. In that case, you can use a planetary gear system, as shown here:
In this gear system, the yellow gear (the sun) engages all three red gears (the planets) simultaneously. All three are attached to a plate (the planet carrier), and they engage the inside of the blue gear (the ring) instead of the outside. Because there are three red gears instead of one, this gear train is extremely rugged. The output shaft is attached to the blue ring gear, and the planet carrier is held stationary -- this gives the same 6:1 gear ratio. You can see a picture of a two-stage planetary gear system on the electric screwdriver page, and a three-stage planetary gear system of the sprinkler page. You also find planetary gear systems inside automatic transmissions.
Another interesting thing about planetary gearsets is that they can produce different gear ratios depending on which gear you use as the input, which gear you use as the output, and which one you hold still. For instance, if the input is the sun gear, and we hold the ring gear stationary and attach the output shaft to the planet carrier, we get a different gear ratio. In this case, the planet carrier and planets orbit the sun gear, so instead of the sun gear having to spin six times for the planet carrier to make it around once, it has to spin seven times. This is because the planet carrier circled the sun gear once in the same direction as it was spinning, subtracting one revolution from the sun gear. So in this case, we get a 7:1 reduction.
You could rearrange things again, and this time hold the sun gear stationary, take the output from the planet carrier and hook the input up to the ring gear. This would give you a 1.17:1 gear reduction. An automatic transmission uses planetary gearsets to create the different gear ratios, using clutches and brake bands to hold different parts of the gearset stationary and change the inputs and outputs.