Filtration has to be the most important part of your tank, well maybe the keeping the water in bit is important. But to keep your fish alive a filter is pretty damn important.
Annoyingly it's also the most difficult to get your head around. There seems to be an almost unending stream of terms and technology to get your head around. Lets try and straighten things out and break it down to its simplest.
Your filter has two jobs, the first to remove solid waste floating in the water. The second is to provide a place for your microorganisms to live. The ones that turn the ammonia from waste to nitrite and then nitrates. That's it. In freshwater, marine is a slightly different ball game.
In general you want your water to go through coarse sponge, through to finer media, then through your biological filtration and then return to the tank. That's it, although you can add extra things if you want to. Some filters are just a coarse sponge, and that does all those jobs in one.
There are different ways to do this. Internal filter, external filter, sponge filter, sump, and undergravel are some of the most popular.
This is where most of us start. These are wonderfully simple and easy to set up. I use them on a lot of my tanks, and they do the job. You can see them as a power head pulling water through the media. That's it.
These are a little more complex, and a lot more adpatable. The water drawn down from the tank due to gravity and pumped back up into the tank. They need to be water tight to stop the water just flooding out as in general gravity is a lot more powerful then the pump send ing the water back up. You can customise these to your hearts content. But in general I have them as coarse media, finer media, biological media.
These are a block of sponge, and you use air to draw the water through and pul the water through the sponge. Cheap simple and a staple of fish houses.
Plates are put across the bottom of the tank and gravel on top. Air stones are placed in the uplifts and they draw water up. Water is pulled through the gravel and up the up lift and the waste is trapped in the gravel.
There will be a longer article on sumps. But these are generally fish tanks that have been customised. The water enters at one end, goes over a series of baffles and then is pumped back into the tank.