----------------------
The filter and what it does for us
Filters are first noted for their ability to keep suspended solids out of the tank wich keeps the water looking clean and clear. The filter-floss or sponge is the "dirt trap" wich cleans the solids from the water as it passes through. This is known as mechanical filtration.
The next thing that filters are so highly prized for is the establishment of a colony of beneficial bacters that process ammonia on down to nitrAtes. This is called bio-filtration. Medias that beneficial bacteria thrive on are sponges, bio-balls, bunched netting, carbon (once full), pea sized chunks of lava rock and some commercial substrates.
The last, but strictly optional, job of the filter is to remove impurities from the water such as medications, phosphates, heavy metals and the like. This is achieved by forcing the waterflow through a chemical media that either adsorbs or absorbs the specific impurity it was designed to remove. Some medias include Phosban, Nitrazorb, carbon and zeolite.
The secondary job of most filters, such as HOB and internal filters, is the oxygenation of the water in the tank. This is acheived by the water return on the filter. In optimal conditions, the filters return would be aimed across the suface of the water. The rippling across the surface causes oxygen to be "injected" into the water and raises the 02 value in the water. There are venturi valves that are used to inject 02. These are commonly seen on powerheads and specialty water returns for canister filters.
It is all of these options and necessities that makes the filter the single most important part of keeping our water clean so that are fish are healthy, grow and are happy.
For a good description of the different types of filtration we normally see on tanks, check this page out: Types of filters
Mechanics of a properly run filter
For most of us, the option of a huge, multi-staged filter like a wet/dry system to keep our water clean is out of the question. We have to resort to using as compact a unit as can handle the bio-load in a given tank. Most ususally, an HOB filter. This forces us to coddle and baby our filters so that none of the viability of the beneficial bacteria is lost due to maintenence. If we didn't do this, our bio-filters would regularly "fail" whenever we cleaned them or changed cartridges. Below, in no particualr order, are a few tips and guidelines to follow:
--Never, ever clean out your tank and filter 100%. Not unless you want to go through the tough task of re-cycling your tank all over again. This goes the same for removing gravel from a tank. If the tank is fully stocked or overcrowded, you can bet that much of the beneficial bacteria is housed in the gravel. Removing all at once causes your filter to have to grow more bacteria to make up for the loss. This could take a week or more for the balance to be met again. Small increments of gravel removal or keeping it in a bag in your tank for a while allows for the colonies to be "transferred" over to the new gravel or the filter.
--Never use water that has not been dechlorinated on your bio-media. The bio-media should allways be lightly swished around in dechlorinated water so that some of the larger chunks of debris come loose and allow better water flow through itself. Same goes for additives like crushed oyster shells or coral and the like. The bags or baskets these are kept in need to be cleaned regularly to provide potimal flowthrough. If at all possible alternate cleaning of different parts of the filter weekly.
--If you have a well established bed of bio-media separate from filter-floss, you can use the sink or garden hose to rinse the filter-floss. I repeat, this can only be done with a system that has a well established bio-filter that is separate from the filter-floss. Again, alternating cleanings helps.
--ALWAYS use a strainer on the intake. ALWAYS keep it very tightly secured. The intake has been the doom to many fish. Let alone the tiny piece of gravel that stops the pump from working and kills the filter. See photo below of the intake strainer.
--Keep the , strainer, uptubes and impeller assembly clean and clear of muck and debris on a weekly basis. Leave any and all buildup of brown gunk (beneficial bacteria) in the filterbox/canister untouched. This is always a plus to have. Unless the filterbox is unundated with goop. then clean it up a bit but not spotlessly. Once again, alternate.
--If you are depending on the "ten times filtration rate per hour" rule of thumb, make well and sure that your filter/s remains on the highest setting (or needed setting) at all times. Also, the water level should be kept at the manufacturers suggested level in the filters manual. At 2 inches below the suggested level, most HOB filters lose a HUGE amount of gph and become waaay less effective.
Another reason for keeping the waterlevel up to the suggested level is because of power outtages. If the power goes out for more than a few seconds, your filter will drain the contents of its basin back into the tank. This might be ok for some brands of filter becasue of the powerful suction they exude, but MANY brands and models will seize up/run dry and overheat.
This also pertains to the oxygenation of the water. If the filter is simply dumping the water back into the tank, it wll create much less oxygenation than if the flow were directed across the surface.
--Look into your filterbox and check the flowrate/waterlevel often. This is the very best way to know exactly when you need to either rinse or change the filter-floss or some other needed maintenence.
--If chemical filtration (carbon, purigen, phosphate removers,etc.) is utilized, regular checking of each type is needed to judge if the efficacy is still there.
--If more than one filter is used on any given tank, try alternating the cleaning and maintenence between the them so that you always have one undisturbed bio-filter going at any given time.
--If you are using a filter that does not have a specified bio-media chamber or the like, when you go to put a new peice of filter-floss in, keep the old one and shove it in behind the new one so that beneficial baceria can more quickly colonize the new floss. The old one can be remove a week or two later or left alone if its not impeding any waterflow through the filter.
------------------------
Below are some photos that show different parts of the filter and whatnot:
This is an exceptionally well colonized bio-wheel from a penguin 330 That has been running for almost 2 years. I NEVER clean a bio-wheel at all unless it stops spinning correctly and the problem is uneven growth.

And here is how you can extend your intake filter to the very bottom of the tank. This results in an amazing difference when you go to perform gravel vacs and theres nothing coming out!

Last but not least, If you are employing a filter that does not have a bio-wheel, media basket or you just want to add some extra space for beneficial bacteria to grow, Sponges work great! You can usually find Aqua-clear sponges at your local fish shops. Just get the right size for your application. Cut the sponge like you would a loaf of bread except do it longways. Cut as thick as you need it for your application. The sponge should sit a little loosely up in front of the cartridge but not wide enough to totally impede the waterflow if it wetre to get clogged. A fingers width of a gap on one side is fine. I see no reason for not adding them in front of the cartridges (really, behind if you look at it from this angle). More space is always better. But, keep them relativelu clear of debris, the same as you would the cartridge/floss. Heres what it looks like:

Here's a link to a very good way to boost your bio-filtration of almost ANY HOB filter: DIY media basket/filter cartridges
As you can see, There really isn't all that much to keeping your filters running properly, it just takes regular observation and a general understanding of the mechanics, and science, involved.
Always remember, a happy filter equals happy water and happy water equals happy fish.
Good luck!