Cycling
The "nitrogen cycle'' (more precisely, the nitrification cycle) is the biological process that converts ammonia (which comes mainly from fish waste) into other, relatively harmless nitrogen compounds. Fortunately, several species of bacteria do this conversion for us. Some species convert ammonia (NH3) to nitrite (N02-), while others convert nitrite to nitrate (NO3-). Thus, cycling the tank refers to the process of establishing bacterial colonies in the filter bed that convert ammonia -> nitrite -> nitrate.
Ammonia and NitrIte are both toxic to fish in relatively small amounts, so this conversion is absolutely necessary. NitrAte is less toxic to the fish, but we still have to maintain the tank and not let the levels get too high. We do this through regular water changes, which essentially dilutes the amount of nitrAtes present in the water.
Most fish-keepers will recommend you "cycling" your tank without fish in it. So build up the bacteria in an empty fish tank (with water obviously) and using the addition of pure ammonia, added by you. This is obviously much safer for the fish and you won't run the risk of any fish dying through the process.
Fishless Cycling
Some newbies however (like myself) find out about the cycling process too late, and have fish in the tank, or at least no viable place to put the fish. Thel in kbelow will give you some information of what is involved with cycling your tank with fish. As you have your fish in a tank already, IMHO you should go ahead with a fishless cycle.
Cycling with Fish
If you hav any other questions about cycling in general, I'd direct them to the "New To The Hobby" section where people can help.
Hope this helps.