the nitrogen cycle

definition
The nitrogen cycle is the most important concept for any aquarium hobbyist to understand, and it's actually pretty handy to understand the basic concepts underlying sewage treatment.

The nitrogen cycle comes in 4 components.

ORGANIC NITROGEN
Poop, leftover food, decaying plant or animal matter.

AMMONIFICATION
Organic nitrogen is decomposed of by bacteria and converted into ammonia. Ammonia is a toxic chemical to life and small amounts of it in the aquarium can kill fish and plants very quickly.

Ammonia (NH3)
Decomposed Organic Nitrogen

NITRIFICATION
Aerobic bacteria convert ammonia into nitrite. Nitrite is still a very hazardous chemical in the aquarium environment, although slightly less toxic than ammonia, it can still cause quick deaths in the aquarium.

Nitrite (NO2)
Aerobic Nitrosomonas bacteria (example)

Other aerobic bacteria then convert nitrite into nitrates. In high enough concentration, nitrate is still capable of killing lifeforms, however most aquarium life can tolerate up to 2 orders of magnitude more than ammonia or nitrite. Typically, most life (except the more sensitive ones) can handle up to 80ppm of nitrate without issue, and with time to adapt animals and plants can tolerate much higher levels of nitrates.

Nitrate (NO3)
Aerobic Nitrobacter bacteria (example)

DENITRIFICATION
Although most aquarists use water changes to lower the level of nitrates within a body of water, one major method used by nature to remove nitrates is the usage of anaerobic bacteria which convert nitrates into nitrogen gas. The nitrogen gas is then released back into the atmosphere.

Nitrogen (N)
Anaerobic bacteria