Filtration

Adequate filtration is essential for all tanks – excess food breaks down, consumed food is broken down by the fish and expelled. This creates ammonia which then needs to be broken down to nitrites which in turn are broken down in to nitrates.
This is known as ‘The Nitrate Cycle’.
 
You will see the final stage mentioned above is nitrates – this is because each stage is less toxic than the last, with fish more tolerant of nitrates than ammonia or nitrite both of which can prove fatal. In fact, nitrates can be broken down further though this is more complicated (this stage is more important in reef tanks where nitrates are actually more toxic in saltwater and fuel nuisance algae that can overrun and kill corals) – In freshwater, it is generally sufficient just to dilute nitrates by way of weekly water changes (maybe 25% per week).
 
Filters create an environment where various pollutants are removed/broken-down.
 
Types of filtration:
 
Mechanical
physical removal of solid waste before it can break    into toxic forms. This can be as simple as a sponge,  though if this is not cleaned regularly it will become a biological filter.
Biological
where filtration media creates an environment allowing    waste to break down (see nitrate cycle above) – usually a material with a high surface area (a sponge is a simple example) allowing the various bacteria to perform their duties.
Chemical
usually by way of adsorption – for example, media which remove phosphates by binding them to itself. The most common example of chemical filtration is the use of Activated Carbon.
 
Types of filter:
 
Undergravel
the aquarium substrate lays on top of a layer of   perforated plastic through which water is drawn by way of either an airpump or preferably a powerhead (this can be done in reverse which reduces the chances of the substrate becoming too compacted. The gravel needs frequent cleaning which is a bit of drawback in that it can be quite messy and disrupt the tank. It’s good point is that it is a very economical method.
Internal
this is basically a box containing media (usually sponge   but chemical media can usually be added) through    which water is forced, usually by way of a powerhead  though simple ones can be air driven. These are efficient for small to medium sized tanks.
External
in this case, the filter is a combined container/pump        that sits remotely from the tank (usually underneath in the cabinet) – these are extremely efficient and are     good on larger tanks. They usually have compartments for various media and some models even have combined heaters.
Hang on Back
there are models that are designed to hang on the back   or side of a tank which reduces clutter in the tank. Often, they incorporate a protein skimmer (see below)  which makes them ideal for converting a smaller tank to a reef environment.
Sumps
these are secondary units (usually kept below the tank   in the cabinet) separated into various compartments to   perform various duties (e.g. protein skimming). These have the advantage that most equipment (except pumps to create currents) can be hidden from view. They also   increase the overall water volume which improves stability.
Protein Skimmers
these are a relatively recent innovation whereby water    is mixed with vast amounts of air in the form of micro-bubbles. The contact point between the water and the   air attracts waste which then ‘collapses’ into a collection cup as the foam/bubbles burst. This can be equated to the foam that accumulates on the seashore.   These are predominately used in marine aquaria though freshwater models do exist (they work differently because bubbles don’t remain small in freshwater long enough).
 
Overall, filtration is such a vast subject we are unable to do it sufficient justice – again, if you are interested, there is plenty of research you can do.