Cycling is the process of establishing colonies of beneficial bacteria in your filter. Fish waste is basically ammonia and is toxic to your fish. Ammonia is kept from killing our fish by developing colonies of these bacteria that convert ammonia to nitrite. Nitrite affects the blood's ability to carry oxygen; so, it can also be toxic. Luckily, a second set of bacteria will colonize and convert nitrite to nitrate. Nitrates aren't generally considered toxic until they reach higher levels. Doing regular, partial water changes, using plants, or both removes nitrates. It usually takes 4-6 weeks to get large enough bacteria colonies to keep up with the ammonia and then nitrite levels. This is why it's always recommended that you start with just a few hardy fish. Feed them small amounts (what they eat in 30 seconds) once a day or even once every other day until your ammonia and nitrite levels have risen, and returned to 0. Then you can slowly start adding fish every few weeks until you get to the expected bio-load for the tank. The term bio-load refers to the maximum amount of ammonia your bacterial colonies will need to process when your tank is fully stocked. In other words, the maximum amount of fish, food, and waste your tank can handle.
It's normal for your tank to initially become cloudy. The beneficial bacteria that digest the fish waste colonize only on the surface areas in your tank. They form in a layer, one deep, on any surface that has oxygenated water passing over it. The cloudiness in the water is an initial bacteria bloom that you'll see as the colonies start to establish. Don't panic and try to get rid of it with chemicals or a UV sterilizer, you'll just slow the process down. Patience is a virtue, but I usually have to remind myself of that as I'm shopping through Ebay looking for any to spare!
Fishless cycling sounds very involved and scientific but it's really not too difficult. One of the biggest challenges can be obtaining pure ammonia. You want to make sure you get a type without foaming agents, colors, or scents. That will usually be the cheapest brand available. Add ammonia to the aquarium to reach a level of 5ppm, and keep track of how much ammonia that takes. Add that same amount daily until you start to see nitrites, then reduce the level to ½ the amount you had been adding. Once both ammonia and nitrite read 0, you can do a large volume water change to reduce nitrates to a manageable level and add your new fish! (Many thanks to Nomad for developing this wonderful time saving method.)
If you are cycling your aquarium with fish, adding Amquel can detoxify the ammonia for your fish, if it reaches levels that would stress your fish. Be sure you don't overfeed during this time. Uneaten food can create an ammonia "spike", which can permanently damage their gills, if your fish manage to survive it. Ammonia will make fish itch, gasp at the top of the water, show inflamed, red gill tissue, and can cause septicemia (red streaks in their fins) along with numerous other systemic problems. Be sure to keep track of your levels! Water changes of 30% will help keep your fish from being as damaged by the levels of ammonia and nitrite, or help you get levels under control if you've discovered you added too much ammonia by accident with the fishless method. If you are doing a fishless cycle, I definitely have to recommend not using Amquel, amrid, or any other additive that is going to interrupt the ammonia availability. You'll risk slowing down the process.
Adding salt at a dosage of 1 teaspoon per gallon to tanks can help your fish handle high nitrite levels. Adding power heads or air stones can also be a very good idea.
There are many available products that claim to contain cultures of the bacteria needed for an established aquarium. In my experience, most of these products do nothing more than shorten the process by a few days, if even that. Bio-Spira, by Marineland, is the only "bacteria in a bottle" that I had really noticeable results using. I definitely recommend it for speeding up the process if it's available near you. Filter media from an established tank, gravel from a tank with an under gravel filter, or even the brown nastiness from a filter cartridge or sponge rinsed in aquarium water will all aid in jumpstarting those bacteria colonies. "Borrowed" media can be placed in a filter mesh bag (no nylons, they can have bacterial agents in them), then added to your new aquarium. Doing this can shorten your waiting period to a matter of days, rather than weeks for the cycling process to complete. (Yes, I know the idea of messing up your beautiful new aquarium is hard to take, but it DOES do wonders!) I personally tend to be into instant gratification so this part of the process is very important!
Good luck with your new aquarium! If you have any questions post them on the Fishaholics.org discussion board and we'll see if we can clear them up for you!