In a nutshell...
Fish pee in the water and it makes
ammonia. Ammonia is bad for fish. Bacteria converts Ammonia into
nitrite. Nitrite, too, is bad for fish. Different bacteria convert Nitrite into
nitrate. Nitrate, too, is bad for fish, but only in high quantity. There are bacteria suppliments on the market to kick start the cycle. IE Biospira, Cycle, StressZyme, etc. Typically you will see the ammonia spike, then the nitrite spike. When they are both at 0 and your nitrate is less than 40, your tank is cycled. Water changes keep nitrate in check and you can monitor these levels with a
test kit or strips. If you test every day you will go nuts and lose patience. Every other day. BioSpira is full of live bacteria that cuts the cycle time from weeks to virtually hours. It is pricey but works. Most fish sites explain the cycle further, but that is the 2 cent answer.