Cycle is the term used to refer to the Nitrogen Cycle that takes place in your tank when you start it up. When you add something that rots or creates waste and creates harmful ammonia it triggers bacteria to start growing that will consume the ammonia and turn it into slightly less harmful but still plenty harmful nitrites, which then another type of bacteria consumes the nitrites and turns it into much less harmful nitrates. This is what a cycle is.. a new tank goes through a Cycle when you first try to establish the bio-filter. Under certain circumstances your tank may go through 'mini-cycles' that arent quite as dramatic as your first main cycle.. but can still stress fish if you already have a established aquarium..
I hope that helps.