Filters are not actually a necessity. With a well balanced, heavily planted tank, low stocking levels, and regular maintenance (which most of us don't have) a filter is not required. It is recommended though, especially for newbies.
Pumps also power under gravel filters, as Jude said, plus box and sponge filters.
Air control valves are also used when you have several filters running off one pump. With different lengths of tubing they need the air regulated so all pump the right amount of air through. They can also be used to slow down the air flow if it is making the filter too noisy with too many bubbles coming through too fast.
The kind of filter you have is personal but can also depend on the type of fish you have. Many fish like to dig in the substrate so you would not use an under gravel filter. Fish producing fry need a filter the fry won't get sucked up into. Fish who do not like strong currents (like discus, bubble nesters etc) also need gentler filtration whereas the ones who live in fast flowing streams need all the current they can get - and a powerhead would help there.
That is an excellent site Caper.