From FrugalReef

The nitrogen cycle is the bacterial cycle by which waste products and decaying organisms in your tank are rendered harmless. Understanding it is very important for any aquarium keeper. Knowledge of the nitrogen cycle can help you avoid the principle pitfall of new reef keepers: New Tank Syndrome.

Ammonia enters the aquarium either as waste from the inhabitants or from the decay of dead organisms. Nitrifying bacteria convert ammonia to Nitrite. Nitrospira bacteria convert Nitrite to Nitrate. Denitrifying bacteria convert Nitrate in the water to gaseouse Nitrate that will bubble out of the tank. Ammonia and Nitrate can also be absorbed by growing plants.

Establishing the bacterial colonies to neutralize this waste is the primary goal of cycling.