jeeeeeez this is complicated
Ah don't worry about it, it looks complex at first, but its really easy to learn once you have got the hang of it.
Basically the main things you need to know with the bacterial cycle is that
Ammonia>nitrites>nitrates
Fish pee and poop ammonia, which incidentally is poisonus to them and can kill them.
There is a bacteria that lives in your filter sponge though that can break down ammonia in to harmless substances, and as long as the bacteria has oxygen (from water being pumped through the sponge it lives on in the filter providing oxygen for it and food i.e. waste sucked up from the tank into the filter) and an ammonia source (i.e. fish poop or rotting matter in the tank), then your bacteria will thrive and get rid of the ammonia, turn it into nitrites (which are also harmful) but eventually into nitrates (which will be absorbed by any live healthy aquarium plants you have in the tank), which are generally harmless.
The process of getting your benneficial bacteria established in your filter sponge is called "cycling", and can take a while for it to finish, but when the tanks ecosystem/bacteria stabilises itself, you should have no ammonia or nitrites and nitrates should be preferably under 40 and your fish will be happy.
As i said though, for this bacteria to survive it needs a constant ammonia source in the tank. You can either fishless cycle the tank by adding pure ammonia to the tank and regually testing the water quality, and add the fish after its cycled, or you can cycle the tank with very small quantities of fish and rely on the ammonia they produce to cycle the tank.
The bennifits of fishless cycling? Because you don't cycle the tank with fish, you don't have to worry about them getting killed off or sick by the levels of ammonia or nitrites being produced in the tank, so you can add large quantities of ammonia (which would normally kill fish) because you don't have fish in the tank, and cycle the very quickly.
It also means that no fish get ill during the cycle, and you can stock many fish in there once it is cycled.
The bennifits of cycling with fish? No many, but it does mean you can stick a couple of fish in the tank straight away, but because you cannot let ammonia or nitrite levels rise much, you have to do lots of water changes and the cycle ends up taking much slower to complete.
You can only use very hardy types of fish as many fish are very sensitive to water quality issues like ammonia or nitrites.
So now days, most people opt for fishless cycling due to it being safer and faster, even if a little bit more complicated than cycling with fish sometimes.
Edit: oh, i forgot to add, if you know anyone with an established fish tank already, you can just stick your new fish tank filter in their tank for a week or two and then transfer it back to your tank- during this time your friends benneficial bacteria would have settled on your filter.
So you can basically cycle your tank very quickly- this method is called "cloning" and when you clone a tank, this is basically what you do, its an easy way to cycle the tank without going through a complicated fishless cycle or lengthy cycle with fish.
The only draw back is that as soon as you turn your filter off to put it back in your tank, you need to get some fish in your tank ASAP as the bacteria will start to die off as soon as the filter is turned off and there is no ammonia source.
You can even use pond filter media to clone your tank with
.
