If you put a box filter in the corner where its most convenient, it becomes a corner filter. Somehow, more people refer to this kind of filters as corner filters than box filters so we'll use the term corner filter here. Corner filters come in different shapes and sizes, but the concept is the same.
1. Air is pumped using an airpump into the corner filter
2. air bubbles rise along a passageway
3. rising air bubbles pushes water out of the passageway
4. water is drawn in to past the filter media contained in the box to replace the displaced water
Corner filters are actually for biological filteration. The wool at the water intake is used only to prevent premature clogging up of the micropores of the corner filter. In deployments where only wool/sponge is used and no porus bio media is used, the wool/sponge serves as the media for beneficial bacteria to grow on. Corner filters are not effective for mechanical filteration as the suction force is not strong enough to move solid wastes.
Corner filters are actually effective filteration provided the bioload it is supporting is small. Due to the limited amount of bio media a box filter can hold, it cannot provide enough surfaces to support a large bioload tank.
The advantage of using box filter is that it provides aeration and filteration at the same time. It is an effective way to provide filteration for a number of small tanks, such as guppys breeding tanks as you only need a strong air pump with air tubes going to all the tanks with corner filters.
Another advantage of corner filters is that they do not have a powerheads. Due to this, corner filters are very useful for tanks(ie shrimp breeding tanks) that need to keep the temperature cool. Different powerheads heat up the water to different extents, but without powerheads, plus the air bubbling, tanks powerered by corner filters are substantially cooler.
One mistake to avoid if you are using corner filters with fully wool is to keep changing the wool once it looks dirty. As mentioned earlier, corner filters are biological filter. When porous bio media is used, the wool actually acts as the bio media. If you throw out the 'dirty' wool, you are throwing out the beneficial bacteria. People who makes this mistake will find that the corner filter is a piece of junk. The reason is simple, the beneficial backteria never got the chance to do their work if they get replaced every time they establish.
The second mistake is that when used to support more bioload than they can handle, the aquarists have to use water changes to complement the filteration. It is essential to use pre-dechlorinated tapwater. If pure tapwater is put into the tank followed by antichlorine, the already limited amount of beneficial bacteria in the tank will be affected. It is essential to pre-dechlorinate the replacement tapwater in a bucket or pail before introducing them into the tank. This procedure is less important with tanks using other kinds of filters with a lot of bio media, because the beneficial bacteria is spreaded out. When some of the beneficial bacteria is killed, other unaffected beneficial bacteria can reproduce to replace them. In a corner filter, all the beneficial bacteria in the limited amount of bio media can get wiped out in an improper water change.
You may have seen fish shops using corner filters for tanks housing fishes that arent exactly small. Remember that for bioload it is not the size of the fish that matters, but the amount of food that goes into the system. Most fish shops feed fish very minimally as fish can actually last more than a few days without food very well (except fish fry). Within that time, the fish would have left she fish shop and been sold to a customer already. Another reason is that due to the large volume of fish passing through the shop, it is very easy to get diseases in the tanks and filters therefore they often have to throw out all the media that tank use. It wouldn't be cost effective to throw out whole sumps of ceremic rings every time outbreaks happen.