We know harmful and beneficial byproducts inhabit our tanks, but do we know how these things come into existence and what causes them? After doing much research, I've compiled this information into a simple format that even the beginner aquarist should comprehend.

The cycle begins when goldfish are placed in the tank. Their poo and uneaten food begin to decompose at the bottom of the tank, as well as any dead or decaying plant pieces. Now in the natural environment, the ratio of fish to water space is much lower than in a small aquarium. Even a very large tank is small compared to a giant lake. So waste products and plants do not in general affect the overall conditions of big bodies of water. In a tank, these products quickly accumulate and destroy our beloved pets. How? Fungi and bacteria convert the waste into something very bad called AMMONIA. Ammonia can be ionized or unionized. The ionized form can be found if the ph is below 7. It is not in any way harmful to fish. However, the unionized form is found if the ph is above 7 and is VERY toxic to fish. Ammonia is said to be the #1 killer of all aquarium fish. Signs of
ammonia poisoning include:
1) Torn, ragged fins
2) Bloody streaks in tail as well as blood hemorrhaging on the body
3) Listless as well as bottom sitting
4) "Flashing" and rubbing on gravel and other objects
5) Gasping at the surface
6) Clamped fins
Yeah, it's that bad. When signs like these are spotted, they are sometimes mistaken for diseases and illnesses. Inexperienced owners often treat with the wrong medications, which only deteriorates water parameters and usually pushes the suffering fish over the edge. When signs like these are spotted it is imperative to test the water first. Ammonia is dangerous if it can be detected using a tester kit. Rule of thumb: If it is detected, it's dangerous. Black patches/spots are often seen as a sign of healing after a bout of ammonia poisoning. To eliminate the awful presence of ammonia from your tank, it is imperative to do regular water changes. 50% weekly is recommended.
Nitrosomonas (a type of bacteria) shows up and begins to eliminate the ammonia. Sounds good, right? Well Nitrosomonas has a bad side effect because it has a byproduct called NITRITE. Nitrite is also very poisonous to fish in very low measurements. Signs of
nitrite poisoning are often similar to those of ammonia poisoning:
1) Bloody streaks in fins
2) Blood hemorrhaging on belly (at this point, it is very likely you cannot save the fish)
3) White patches (burns) on body
4) Clamped fins
5) Flashing
To counter the effects of nitrite poisoning, it is helpful to do a large water change and add a .3 concentration of salt, which absorbs nitrites around the gills. But an ounce of prevention is worth a pound of cure, so it is better to keep the tank clean regularly than to stress later when your fish come down suffering.
Nitrobacter (another bacteria) comes along and turns the nitrites into NITRATES. Nitrate is harmless to fish unless it is found in quantities of over 40ppm. Then it causes rapid oxygen depletion, making it unable for fish to breathe. Nitrates are useful because live plants in the aquarium can feed off of them. Once nitrates start showing up on your water test, it means your aquarium is cycled.

Cycling the aquarium provides a constant parameter monitor, and in turn provides you with healthy fish.
Understanding the nitrogen cycle can help you to be the best fish owner possible.