The nitrogen cycle

It's a big problem for many beginner aquarists. The fish and other little critters you may have produce ammonia, which good bacteria convert to nitrites, which are then converted to nitrates. Nitrates are nowhere near as bad for you fish as ammonia (or ammonium, depending on the pH of your tank) and nitrites. The bacteria have to build up over time; you need to "cycle" a new tank with a source of ammonia for usually about four weeks for a good amount. If you can get any material (filter media, substrate, decorations) from an established tank, it's called "seeding" the tank, and the cycle will go much faster and you can get live plants and animals quicker. A good source of ammonia for a new tank cycle would be fish food, as broken-down food gives off ammonia like fish waste. Two types of bacteria, Nitrosomonas and Nitrospira, make up the cycle.

1. the fishless cycle: You set up your brand new tank and put all your substrate and decorations and everything. Install the heater (if needed) and filter, then fill the tank with fresh dechlorinated water. Plug in your appliances; use a thermometer to make sure the heater works and keeps the water at a stable temperature (usually 76-80F is good for a tropical community). If you can't get any material from an established tank, then throw some fish food in every day and test the water for ammonia. 
   
    You should get an ammonia spike after a week or two, and should then start testing for nitrites but keep up with the food. After about another week or two, the nitrites should spike and start to go down again. Once the ammonia and nitrites are completely gone and you have a reading for nitrates, you can start getting fish and plants. Only get one or two fish each week so you don't stress out your bacteria culture. Make sure to research the fish and plants you buy to make sure you can give them the proper care and see if they are compatible for your tank's environment and other inhabitants of your tank.
   
   
   
2. cycling with fish: Another option is to use a fish to help cycle the tank. Since this option may cause serious illness or death to a live animal, it is not recommended. Set up your tank and let the appliances run for three days to make sure they're stable. Then go buy one very hardy fish, like a betta or zebra danio (GloFish are cooler than plain zebras though) and introduce it to your tank. Only feed a few flakes (or whatever it eats) a day, since the fish will produce as much ammonia as you need for the cycle; any extra food is just more for your new bacteria to process. If the fish dies during the cycle, examine the body to make sure it wasn't ill; if it seemed healthy, leave it in the tank since it still produces ammonia which you're going to need for the cycle to finish. 
   
    You should get an ammonia spike after a few days, and should then start testing for nitrites. After about another few days, the nitrites should spike and start to go down again. If the fish died, you can remove the decomposing body and change 25% of the water. Once the ammonia and nitrites are completely gone and you have a reading for nitrates, you can start getting other fish and plants. Only get one or two fish each week so you don't stress out your bacteria culture. Make sure to research the fish and plants you buy to make sure you can give them the proper care and see if they are compatible for your tank's environment and other inhabitants of your tank.
   
   
   
3. bacteria cultures: Instead of using either option given above, you may decide to try a bottled bacteria culture. Using Colonize has given many people success with seeding their tanks. Another good product is Tetra SafeStart, which has given some success.

To keep your fishy friends healthy and happy, you should do partial water changes each week. Take about 20-25% of the existing water (use a gravel vacuum to remove waste and leftover food from the tank) and replace with fresh dechlorinated water. Make sure the temperature of the new water is the same as the old water. This helps to keep nitrates down and replenish trace minerals in your aquarium. Plants also help with nitrates by absorbing them and using excess food and fish waste as fertilizer. When changing filter media, only change a little at a time; like if your filter has a sponge and carbon and floss, change the sponge one week, the carbon the next week, and floss the week after.

(Is it bad that I learned more chemistry stuff in fishkeeping experiences than I did in my chemistry class last year?)