When we speak of "Cycle" we are talking about a nitrogen cycle.  The food rots, fish waste, Bacteria die ect this is a natural part of a life cycle...  However, the byproduct of this event is Ammonia.  Ammonia is highly toxic to all living creatures..  In a "fully cycled" system we have at least two forms of bacterial colonies that when fully established will break these harmful compounds down into less and less toxic forms.

So here is a run down of the "CYCLE" in a healthy system

WASTE  =  very very bad, but unavoidable

to

AMMONIA  = very bad, but simply a byproduct of life

to

NiTRIfying bacterias will now take the ammonia and digest it into "Nitrite"

to

NiTRAfying bacterias will now take the ammonia and digest it into "Nitrate"  This is the least toxic form of waste and represents the end of this cycle

These bacterial colonies do not yet exist in a new system or are intially far too weak to handle the bioloads of even thier own cellular respiration.  For this reason we "CYCLE" a tank before adding livestock.  That is to say we genesis the colonies with live sand or live rock...  even more recently with bottled cultures and best of all "System cloning"...  Then perhaps a cocktail shrimp to rot in the water for 3-4 weeks..  as these colonies exercise and grow they become more and more capable of breaking down the ever increasing bioloads that a closed reef system represents.