In a cycled fish tank there are two colonies of bacteria. The first type 'eats' the ammonia which is fish waste. They turn it into nitrite. The second type of bacteria 'eat' nitrite and turn it into nitrate. Both ammonia and nitrite are poisonous to fish, the levels must be kept at zero. Nitrate is as poisonous as the other two but it must be kept below 20 ppm by water changes.
Cycling is the process of growing these bacteria. From starting a brand new tank it takes several weeks to grow enough bacteria to support a tankful of fish. The process can be speeded up by using filter media from a tank that has been running for more than 6 months, or by adding one of the 'bacteria in a bottle' products.
There are two ways to cycle a tank - before fish are bought by adding ammonia from a bottle to get the bacteria to grow, then get fish when they have grown, this is called fishless cycling. Or by adding a few fish straight away, doing water changes every time ammonia or nitrite show at more than zero.
Once the ammonia and nitrite levels stay at zero, water changes of at least 50% every week need to be done to remove the nitrate which builds up, and all the other things excreted by fish which we can't test for.