Overhead filters works by situating the filter media above the tank water surface. Water is pumped up to above the filter media using a water pump or powerhead, which then flows pass the filter media and re-enters the tank by gravity.
There are 2 very big advantages of using overhead filters. Firstly, beneficial bacteria are able to work very effectively due to the excellent air/water interaction at the filter media. Secondly, it is very easy to do maintainance. The lid can be opened and the wool replaced in seconds without turning off the power supply. Uneaten food and solid fish waste can be easily and regularly removed before they can break down, reducing the bioload on the system.
There is enough gas exchange between the water and air at the water inlet and outlet zones such that aeration with air pump is not necessary and in fact redundent in most situations where the tank is not being seriously overstocked. Water reentering the tank from the top also effects good water circulation in bringing water with dissolved oxygen from the water surface down to lower regions.
The disadvantage of the overhead filter is that the pump take up space inside the aquarium although the media is above the aquarium. The water pump and tube channeling water upwards can also look unsightly and visually obtrusive. (There are some types of overhead filters that have the pump above the water surface where water is drawn up using a vertical pipe, but be sure to get one from a reputable brand if you get this type. There has been real and documented cases where they overheat and cause fires, even burning down the house of some victims.)
Another disadvantage is that should the connection between the water tube and the filter box be loosened and disconnected, the tube could actually start channeling water outside the filter box onto the floor. If your water pump is placed on a low position inside the tank, most of the water could be drained out and cause a minor floor on the floor besides the impact on your aquatic livestock.