The term 'cycling' refers to the build up of essential bacteria, which break down wastes.
In a new aquarium, there are not enough of the bacteria to cope with the waste load and toxic ammonia can rise to dangerous levels. Eventually, the bacteria increase to cope with the ammonia, converting it to another, only slighly less toxic compound - Nitrite. This too will then rise to high levels until a second type of bacteria increases and converts it to the much less toxic Nitrate. This process can take several weeks. In the confines of an aquarium, there is not really a complete 'cycle', and most of the end product, Nitrate, is normally removed by water changes.
There are therefore certain things which should be done to minimise any stress or even fatalities during the cycling period:
Add only a few hardy fish at first and feed lightly to minimise wastes.
Test water regularly and perform water changes to reduce the levels of ammonia and nitrite if they become dangerously high. The bacteria are attached to surfaces, so removing water should not slow down the maturing process.
If possible, obtain some gravel, tank decor, plants or filter media from a mature tank. This will introduce some of the necessary bacteria and may reduce or even eliminate cycling time.
The aquarium is considered 'mature' when ammonia and nitrite have reduced to zero, and nitrates have begun to rise. At this point it will be necessary to begin a regular programme of water changes to keep the level of nitrates low (aim for less than 50 mg/l, less than 25 mg/l is better).
(Note: mg/l = milligrams per litre and is essentially the same as ppm = parts per million, for most purposes).