There are three types of filtration: biological, mechanical and chemical. This site is an introductory site for fish keeping and a description of these filtration types is beyond its scope.

However, filtration is a subject of importance. In short you will need some sort of filtration to get rid of fish waste and uneaten food. Without filtration the water will become polluted with these things and constant water changes will be needed. The other option is if your fish are in an aquarium filled with live plants. In this case the plants will take care of the filtration.

Plants are expensive and difficult to keep alive for beginners since they require an understanding of soil and gravel types and a more detailed explanation and understanding of the nitrogen cycle.

So you will most likely be needing a filter of some sorts. There are two main types. Those which are placed in your aquarium, and those which are outside of the aquarium.

The ones which are placed within are small and usually are made partly out of sponges or a mixture of charcoal and cotton. They work with an airstone and pump which circulate water through the filtration material. The advantages of these are they are small and cheap. However, they require a little more work to clean.

The other type are filters which do not sit in the aquarium, but either hang on the side, or sit on the ground. These use pumps to pump water through their filtration materials and pour the water back into the aquarium. These are easier to clean and filter more material, however, they do cost more.

Whichever type of filter you get you should get something which will filter all the water several times an hour. So if you have a 10 gallon tank then a filter which does 80 gallons an hour is good enough. However that filter will have a problem keeping up with 3 goldfish in a 30 gallon tank.