I can give it a shot.
Fish produce ammonia, as does decaying material. Ammonia is lethal to fish, so something has to make it non-lethal.
The "cycle" people speak of is the cycle that transforms ammonia into nitrates, generally in two steps. In basic terms,
Ammonia (gets oxidized to)--> Nitrites (which gets oxidized to) --> Nitrates
Both of these steps are perfomed by bacteria. Both ammonia and nitrites are considered lethal to most fish, while nitrates are generally considered harmful only in large concentrations.
Conventional wisdom says that the Nitrosomonas species of bacteria handles the first step (oxidizing ammonia into [edit: typo] nitrites), and Nitrobacter species handles the second step (nitrites to nitrates).
However, some fairly recent research shows that Nitrosomonas may not be prevalent in aquaria, and Nitrosococcus mobilis has been mentioned as a possible "substitute", that actually does the job.
Similarly, some evidence shows that Nitrobacter is not prevalent in tanks, and one suspected "substitute" is Nitrospira. -- By the way, this is one type of bacteria found in, and the basis behind, the BioSpira product by Marineland Labs.
Whatever the "truth", one type of bacteria oxidizes ammonia into nitrates, and another oxidizes the nitrites into nitrates. When people speak of "cycling", they usually mean growing a bed of the two types of bacteria needed, in sufficient quantity to process the "normal" amount of ammonia produced in your aquarium.
