Basically, there are three different types of filtration:
Mechanical filtration
is usually done by passing the water through a medium, such as foam or floss, that will trap large particles of debris. Box filters, corner filers, power filters, and canister filters all usually contain some form of mechanical filtration.
Chemical filtration
is a method of chemically purifying the tank water. The most common type of media used is carbon, although other chemicals and combinations of chemicals are also used. The media is either loose or packed into a little nylon bag and placed inside the filter. Box filters and corner filters usually use loose carbon, while power filters usually have carbon in bags. Canister filters, depending on the brand, may use either. Some undergravel filters have carbon-filled caps on the uplift tubes, but these hold only a very small amount of carbon.
What kind of filter is best for MY tank?
The most popular type of filters at the present are external power filters, the kind that hang on the back of the tank. My favorite is AquaClear, but there are many good brands. They combine mechanical, chemical, and biological filtration, are quiet, are easy to maintain, and are relatively inexpensive. I usually get a filter one size bigger than what the manufacturer recommends for the size tank I am setting up. All my tanks have a power filter and an undergravel filter. I would not recommend using an undergravel filter by itself, as it mostly just provides biological filtration and very little else.
Undergravel filters are large plates that must be placed BENEATH the gravel, with tubes that rise up in the back. Something must be purchased to run the filter. Air pumps are inexpensive and can also be used to create interesting bubble effects, but be sure to get one powerful enough for multiple attachments. Airline tubing must be purchased to connect the pump to the airstone in the uplift tube. Another power source is a powerhead, which is usually quieter and not much more expensive.
Box filters and corners filters are very cheap, but are only really good for a tank with one or two fish, or for a breeding tank, where very gentle filtration is required.
Canister filters are excellent, as you can fill them with whatever media you want and in this way can customize them to your tank's needs, but they are expensive, and most beginners find them confusing. They can also be difficult to clean.
For most basic freshwater set-ups, I would recommend a good power filter in conjunction with an undergravel filter run by either an air pump or a powerhead.
