Cycling means that you set up the aquarium with substrate, filter and water, but ideally without animals. After a few days, bacteria will start to produce ammonium out of organic compounds in the water. Later on, other bacteria species will transform the ammonium to nitrite. The last step is performed by a third group of bacteria that change the nitrites into nitrate. Ammonium (NH3) and nitrIte (No2) are highly toxic to aquatic life; nitrAte (NO3) is much less toxic and can be controled by regular water changes.
A tank is considered cycled as soon as there is no more measurable NH3 and NO2 present and NO3 is built up. Before this state is reached, first ammonium and nitrite will spike up until the so-called peak is over. After that, the system is safe for the animals. The time that is needed for this process differs from tank to tank, so it is a good idea to monitor the above-mentioned values on a daily basis with a liquid test kit. Usually, it takes about 4-6 weeks.
There are many homepages that describe the cycling process in more detail, maybe you are interested in reading further on this topic.