I. Nitrogen Cycle
"Cycling" your aquarium refers to the establishment of a biological filter to manage waste. The most commonly found problems with new aquariums, including "New Tank Syndrome," are almost always related to the "Nitrogen Cycle". This cycle breaks down the Ammonia(NH3) found in waste into NO2 and then into NO3.
NH3 is created when fish consume food and O2. The waste is excreted in the form of CO2 which decays into Ammonia. The presence of ammonia triggers the grown of Nitrosomonas which produce nitrites (NO2). The nitrites cause the growth of nitrobactera to produce nitrates(NO3). NO2 and Ammonia are harmful to fish in high levels. If ammonia is not allowed to break down all the way into NO3 then the toxicity of the water will be too high and the fish will die.
This is the detailed step by step diagram of the nitrogen cycle found on wikipedia.
1. Addition of food and nutrients
2. Production of Urea and Ammonia by Fish
3. Ammonia is converted to Nitrites by beneficial Nitrosomonas bacteria
4. Nitrites are converted to Nitrates by beneficial Nitrospira bacteria
...- Less toxic Nitrates are removed by plants and periodic water changes.
7. Oxygen Cycle
8. CO2 produced by Fish
9. O2 produced by plants
II. Cycling Methods
i. Fishless Cycle
The fishless cycle is the most fish friendly method because it does not use fish to cycle the tank at the risk of the fish's life. This method executed by adding raw Ammonia to the tank to provide the begging component for the nitrogen cycle. Then you wait 4-6 weeks while periodically adding more Ammonia to the cycle. Methods to speed up this process include adding existing filter material, water or gravel from an already cycled tank to aquarium in some way. A product exist called "Bio-Spira" but it is very uncommon and only sold in specialty stores. I myself have never seen it for sale so I wouldn't say this is something to depend on.
ii. "Disposable" Fish Cycle
The disposable fish cycle uses a fish to slowly create ammonia for the cycle. Only one or two fish should be added to a 20-30 gallon aquarium on day one. Since there is no existing cycle to break down the ammonia for the first few weeks there will be no ammonia break down. This may cause permanent damage or death to the fish used to cycle the water. Strong fish are suggested for this mission. Commonly used fish are tetras, barbs and dianos because they are usually hardy fish. Anglefish and other delicate or expensive fish should not be used for cycling because they will almost certainly die. This method also takes around 4 to 6 weeks and adding material from existing cycled aquariums can also be added to this process to speed up the cycle and improve the chances of your fish surviving.