The most common variety of "pest" snail is the Ramshorn snail. Apple snails (Ampullaria sp), which can grow to the size of a small grapefruit, are often purposely introduced as part of the aquarium display, whilst Trumpet snails, with their characteristic "cornet" shaped shell, burrow through the gravel turning it over, introducing oxygen and preventing wastes from clogging it. Unlike their more common relatives the Ramshorn snail and Pond snail, the Trumpet snail does not harm plants and is often welcomed by the fish keeper as a sign of a healthy aquarium. The exception to this is the Wandering Snail (lymeaea ovata peregrai) which produces a poisonous substance that can cause convulsions in fish.
How do they get there?
Snails are usually accidentally introduced into aquariums when new plants are added; their jelly-like eggs are attached to the leaves of the aquatic plants.
Snails can be prevented from entering the tank on plants by bathing the plants in Potassium permanganate (available from pharmacists, use just enough crytals to turn the water pale pink), or a commercial snail killer for a few hours, although once introduced, snails can be removed from the aquarium by a number of means.
Note: Potassium permanganate is a strong oxidizer, and can cause burns to any area of contact. It is harmful if swallowed or inhaled.
Why do they stay?
Snails thrive in an aquarium environment because there is a regular food supply. Over-feeding is often a problem and if your snail population is increasing, you need to decrease the amount you feed. This will not harm your fish, as the increased snail population is a sign that you are already feeding more than necessary.
Are they beneficial?
Snails forage on left over food and graze on algae therefore a small colony should not be of concern. In fact, they are doing you a favour by eating excess food (and can be amusing to watch and even add to a more "natural" look in your aquarium!). Some species also help by burrowing through the substrate thus preventing compacting and dead spots and even to help dispose of dead fish. However, as with all living creatures, snails produce excreta and thus large colonies of snails can result in quickly deteriorating water quality. Some species do damage plants, and large numbers may look unsightly.
How to remove them
There are various methods of removing them, either 'biologically', physically or chemically.
If they are suited to the set-up, the best and most natural way is to add snail-eating fish. The best candidates are usually loaches. Clown loaches are one of the most popular snail eating fish, and usually do a good job. If your tank is not large enough for these (recommend 3ft minimum), the smaller Pakistan or Zebra loach may be more suitable. Certain catfish like 'Dorids' (talking catfish) or banjo catfish will also eat snails.
Even if physical removal daily can never completely wipe them out, this is a good way of keeping the population down. "Baiting" often works - if you place a slice of cucumber or lettuce in the tank at night (weighted down so that it stays on the substrate), the snails will congregate on it and then you can just pull them out of the tank with the cucumber slice. One way to avoid the fish eating the slice is to stick it inside a clean bottle, or beneath an inverted plate.
The use of any of the available chemical products is not generally recommended because anything that can kill a snail may also be harmful to your fish and plants. Adding chemicals to your tank is always a risky thing unless you know exactly what you are adding and exactly what the effects will be. Most of these snail-killing chemicals use high levels of copper. A result of this method is the massive die off of snails and the resulting decaying of their bodies. High ammonia levels are the most likely result of this method, so be sure to follow up the treatment with a partial water change. It may be wise to continue with at least 10% every other day for a week or more and make sure to check the filter often during this time - daily monitoring with an ammonia and nitrite test kit after such a treatment is also suggested.
Clearly it would be best to physically remove as many snails as possible before treating with a chemical killer.