dear god, i hope none of you people ever plan to become doctors when you grow up.
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Cycling is the process by which beneficial bacteria establish themselves in your tank. Two kinds of bacteria are needed to convert fish wastes into nitrates. One kind of bacteria converts ammonia into nitrite, and another type converts nitrite into nitrAte. These bacteria reside on SURFACES in the tank - gravel, decorations, glass, plants, filters. They need oxygen to survive, so the areas of the tank with the highest water flow (highest oxygen) will have the largest numbers of bacteria. The bacteria are NOT free-floating in the water, they are attached to things in the tank.
Fish produce ammonia. This is the same chemical used in cleaning supplies. Ammonia harms fish. A level of .25ppm ammonia will begin to burn your fish's gills and irritate their skin. The same for nitrite. Nitrite interferes with oxygen exchange and makes it difficult for your fish to breathe. During a cycle, you should test your water regularly, and DO WATER CHANGES, as often as necessary and as large as necessary, in order to keep ammonia and nitrite under .25ppm. If you are not an idiot and you know how to do water changes correctly - match the temperature, use dechlorinator, don't run filters when adding tap water - even a 90% water change will have NO EFFECT on your bacterial colonies. If, for example, you test your water in the morning and you have 1 ppm ammonia, you should do either one large 75% change, or one 50% change, refill the tank, and another 50% change right after. That will reduce ammonia down to a more reasonable and safer level. If ammonia is .25ppm and nitrite is .50ppm, do a 50-60% change in order to lower the toxins. Once ammonia and nitrite readings remain at 0, and you get a positive reading for nitrates, your tank has cycled.
Trace amounts of ammonia and nitrite (between 0 and .25) are enough for bacterial colonies to grow and complete the cycle of the tank. While a fishless cycle using bottled ammonia in high doses (3-5ppm) will make for a faster cycle and less work, a fishy or fish-in cycle needs to be overseen and managed in order to keep the fish healthy. Ammonia and nitrite poisoning during a cycle, even if it does not kill a fish during that time, can harm the immune system and gills and cause the fish to be weaker and more susceptible to illness and stress for the duration of its life.