A "corner filter" is a box placed inside the tank and filled with filter floss or cartridges. A pump forces water through, mechanically filtering out solid wastes, and bacteria can set up housekeeping in the medium to provide some biological filtration too.

A "canister filter" is a more powerful version of the corner filter. It can be placed inside, or hang outside or underneath the aquarium, and can handle really messy fish, but you do have to clean it very frequently. Like corner filters, you can get a good colony of bacteria to settle in this type too.

A "fluidized bed filter" pumps water through a mechanical and chemical filter, then through a tube full of sand which gives bacteria lots of places to grow. It works best with very oxygen-rich water.

A "power filter" hangs off the back of the aquarium, providing easy access for cleaning and changing the filter medium. It pulls water out of the tank to be pumped through chambers for mechanical and chemical filtration, and many are set up to provide ample space for bacterial colonies to set up biological filtration too.

A "sponge filter" is a tube filled with a spongy medium for bacteria to colonize. The sponge filters water mechanically while the bacteria do their biological bit. Some newer types use two sponges, so you can change one at a time, always leaving one active colony in the tank.

An "undergravel filter" or "UGF" is a plastic plate that you put in underneath your gravel, which lets water or air bubbles trickle slowly up through the gravel to aid bacteria in colonizing the gravel. This doesn't really do any mechanical filtering, and might have to be combined with another type of filter to remove solid wastes. This type also clogs frequently, and may need to be replaced after a couple of years of use.

A "wet-dry filter", or "trickle filter" helps to increase the surface area that can be colonized by good bacteria. It is usually made up of a dry plastic tube filled with a filter medium (floss, sponge, paper, etc.), and your tank water is drawn up to trickle down over the medium. This gives bacteria a much larger air to water surface, and thus more oxygen.

All you need to do is find the best combination of filters for your particular livestock, keep the oxygen flowing to keep your bacteria alive and functioning well, and keep the filters clean, and you'll have a lovely tank and happy fish.