Air Operated
These are filters that sit in the tank itself. Besides being unsightly, it takes-up precious space in the aquarium. Most of them use a filtering media of sponge with or without activated carbon or just filtering wool. 

At the bottom end of filters, are the Air operated type of filters that come in many different shape and sizes. You use filter wool stuffed into a compartment and water is drawn through the filter wool and up the pipe with the air bubbles (uplift). All their functions are similar. Although effective in its own right, its success is limited to the immediate surrounding area and power of the air pump.
As you can guess, the effectiveness is rather limited and depends largely on the air pump itself. These filters are suitable for small aquariums, as most powered types will be too powerful for them.

Powered internal filters almost always use only a sponge. Manufacturers will have you believe that the sponge will grow good bacteria, etc., but such filters can only work as a mechanical filter only. As the surface area is pretty small, it chokes up fast and each time you wash the filter, any bacteria present will killed or be washed away. Forget about using them as a biological system. The principal of operation is the same as the airlift filter box where water is drawn through the filtering media (no matter what it is) and out through a spout. Most effective for free-floating dirt in the water. Change the sponge after about 6 months or if you find the sponge becoming inelastic. Nowadays, these filters are relatively cheap. 

Problems with these filters are that each time you remove them to wash, dirt is bound to flow back into the tank. Making your tank looking worse than it does.

Air powered external filters have very limited effectiveness 
They were popular once years ago, and have fallen out of favour since.

Mechanical & Chemical Filtration possible.

Very powerful pump output and low wattage but expensive. The large canister allows for different filtering mediums to be used. Mechanical, Chemical & Biological Filtration possible. They canister can be hidden in the fish tank cabinet.
A canister is placed outside the tank and water is drawn-in through a pipe into the canister through the filtering media (Filter wool, carbon, etc.), through the motor head and back into the tank. Suitable for large fishes like Goldfishes, Kois, Arrowanas, etc. When you fix your external filters, please check that the pipes are not kinked or the flow will be restricted putting stress on the pumphead. With these filters, you have a large canister for you to put the filtering media of your choice although the manufacturer mostly almost always provides it.

All you have are two pipes - An inlet pipe and an output jet pipe or spray - in your tank. These filters are very much more efficient as you can have all 3 types of filtering system present - Mechanical, Chemical, and/or Biological. As the filtering output is very much higher, you also get a better circulation of water.

Another type of external power filter is what we call Overhead Power Filter (some call it Gravity Filters). A filter box is place on top of the tank and a power head is used to pump water up into the filter box and the water flows through the filtering media and back into the tank. Personally, I dislike the way the dirty water flows back into the tank whenever the filter wool is removed for cleaning. Most suitable for 24" long tanks only. 

Overhead Filters are neat and can be hidden.
Design is compact 
Good for upto 24” Tank only although 36" models are available.
Dirty water tend to drain back when cleaning. 
Relatively Cheap

Just how effective is this and what is it anyway? The most common Biological filter in use is the Undergravel Filter, or sometimes called Sub-sand filter. A porous filtering plate, usually made of plastic, is filled with sand or gravel to about 3" thick. Air bubbles lift water up the airlift tube. Water is forced by natural forces to flow through the gravel/sand, to the undergravel plate, under the filter plate and up to the uplift tube. 

Commercially available powerheads can be adapted to fit on the up-lift tube, making it even more effective. For this filter to be most effective, the plate must cover at least 60% of the base. 100% is most ideal. With the gravel or sand acting as the filtering media, your whole tank becomes the filter. The gravel or sand is expected to be dirty over time. What happens here is that bacteria will grow in the gravel or sand and convert the ammonia & nitrites to nitrates (chemical filtration), which can then be removed by the live plants or by water changes. Large dirt particles will be trapped at the lowest point of the gravel and should be constantly removed via weekly/fortnightly or monthly partial water changes.

At least once, feed your fish with tubifex worms. Place a small bowl or dish in the tank and place cleaned worms in the dish. Spread a small lot all over the tank (Do NOT OVER DO THIS). The worms will go into the gravel and help keep the gravel from choking-up - just like earthworms in the garden soil. Should they manage to reach the plate, they will be pulled out through the airlift tube and become a nice treat for the fishes. 

Important note: the gravel or sand must NOT be coral sand, crush corals, etc., unless your setup is Marine or for Hard Alkaline water fishes. 

If you are using crushed corals, as water flows through the sand at a constant rate, it will dissolve part of the coral, and the water will become harder and harder over time. The pH will also increase along with it. For this reason, if you do not intend to change water regularly, DO NOT USE Crush Corals, Shell, or shell grit in your filtering tank. This is the most common problem in large Koi Pond set-up. Because the tank size is huge, water is hardly ever changed. Overtime, the water gets too hard and alkaline and the Kois or Goldfishes become stressed. Tell tale signs will be revealed later when I touch on Preventive Medicine. 

After setting up an undergravel filter, your tank water MUST be crystal clear in 12 hours. If not, your filter is not working. Your pump may not be working properly. 

Good gravel/pebble size is between 3-5mm. Take a sample of the gravel/sand, place in a glass container, pour vinegar or other forms of acid. If there is
strong reaction - effervescence - do not use for acidic fishes. Slight or no reaction. Its OKAY to use. 

If you do not intend to use undergravel filter system, it is better not to use any sand/gravel at all. 

Advantages of using an undergravel fiter:

Cheap & cost effective 
Cannot be used if fish digs. 
Needs 2 weeks for bacteria to grow. 
Gravel type very important. 
Can use just an air pump or pumphead.
Safe even if there are baby fishes.

Nowadays, most filters make use of a combination of two or more of these systems. Using a filter wool, activated carbon and ceramics in the filtering chambers is an example of a combination of all 3 systems in action.

The problem with any purely biological system is that large dirt particles take a long time to be broken down. So the Best system is one where all three system is at work. Using an external power filter to act as the mechanical and chemical filter while having undergravel to do the biological work. This is very effective for a marine set-up as well. Of course if you have fishes that digs alot, your filter system will go haywire. Either use a larger pebbles or place a filtering wool over the plate. 

In a marine set-up, place a thin layer of filtering wool over the plate to prevent fishes or other creatures from exposing the plates. In such set-ups, normally, fine sand is used instead of gravel. The wool will also prevent the finer sand particles from falling through.

For small tanks using the box filters, you can modify such filters to be biological filters by simply substituting the wool for freshwater gravel. When water passes through the gravel, bacteria will grow. Because the actual filtering surface area is small, the amount of bacteria that grow will be very limited. That is to say, the efficiency will be limited.