Here is the nitrogen cycle in a nutshell:
Fish poop/waste (nitrogen)=ammonia which is very bad/poisonous for the fish. In an aquarium, it has no where to go, so the fish get sick. But, water changes can take it out to a certain extent. But most hobbyists don't want to do 100% water changes daily--especially not on big tanks. Luckily, good bacteria will come along (eventually) and turn the ammonia into nitrite.
nitrite is also very bad for the fish, but not quite as bad as ammonia. However fish still cannot tolerate any nitrite in their tank. So, even with good bacteria number one working, massive water changes are needed. But, then good bacteria number two comes along and turns the nitrite into nitrate. Nitrates are MUCH better for fish. Too many nitrates are not good, so every once in a while (weekly usually, maybe every other week if you don't have too many fish in the tank, or you have lots of live plants that are healthy and growing), you need to change some water. But, fish can live and be happy and healthy with some nitrates in the water, as long as there aren't too many. So, once a tank is cycled, it has two types of good bacteria. One turns ammonia into nitrites, and the other turns nitrites into nitrates. The responsible fish keeper then does water changes so the nitrates are not too high. A well maintained fully cycled fish tank should have 0 ammonia, 0 nitrites and under 40ppm nitrates (ideally under 20ppm nitrates). You can test the water and see what is in it with a test kit. The API master freshwater test kit is the most popular one around here. It is a liquid test kit, and comes with instructions on how to use it properly. It is most important to test while the tank is cycling. Then, daily testing is needed. Once the tank is established, you may find you only need to test if the fish seem sick, or every once in a while for peace of mind.
Most pet stores also offer free water testing. But, depending on the store, you get what you pay for. The big box stores usually use dip strips, which aren't as good as the liquid kits, and the person who tests your water there might not know how to interpret the test as well as one would like.
Emily