Filtration can be a touchy subject, as just about everyone has his/her own ideas on what constitutes a "good" filter. May I suggest we try to establish what the filtration system should do (and not do...), and then look at options. I think we will all agree that the water quality of our tanks must be as close to that of natural seawater as possible, and that proper filtration plays a vital role in achieving this. I offer this "Introduction to Filters 101" as a starting point for the beginner's own research (I definitely do not presume to have all the answers, in fact, I still have many questions myself, so don't take this info as Gospel).
All living organisms eat some kind of food, and excrete some kind of waste. In our tanks this waste accumulates as "Dissolved Organic Content (DOC)", for want of a better word (sorry, I'm not a biologist, and must express myself in layman's terms). The DOC breaks down to ammonia, which is deadly poisonous to fish and inverts, even at very low concentrations. Certain "aerobic" bacteria break down this ammonia to nitrites, which is slightly less poisonous. Other "aerobic" bacteria then breaks down the nitrites, and turn it into nitrates. (These "aerobic" bacteria require oxygen rich water in which to live). Nitrate is much less poisonous - fish can tolerate up to 60 - 100 PPM, and inverts up to 5 - 20 PPM, I've read. However, "tolerate" is far from "enjoy", and we should strive to reduce the nitrate level to not more than 20 PPM for fish, and less than 2.5 PPM for inverts (again, in my opinion). The nitrates are further "broken down" to harmless nitrogen gas by "anaerobic" bacteria, which only live in an oxygen poor environment.
They have been the main "biological" filters for as long as I can remember. They supply oxygen-rich water, and a large surface area, allowing aerobic bacteria to thrive. They do not allow any anoxic conditions, so there is no nitrate reducing bacteria. They can thus lower ammonia and nitrite levels, but will increase nitrate levels in the long run. Also, waste is drawn into the sand, where it can accumulate and block the free flow of water, thus reducing the efficiency or the aerobic filtration as well. They require frequent cleaning, but are difficult to clean (siphon "vacuuming"). They can only survive being "switched off" for a short time (2-4 hours) before the aerobic bacteria start to die off. Used with "power heads" they supply excellent circulation.
Another "old technology" mainstay. They are the most efficient removers of water borne particulate waste. Unfortunately they do not really "remove" the waste from the system, they only concentrate it inside the canister. They therefore require very frequent (like daily) cleaning to remove this waste before it decomposes, releasing the DOC back into the water. If used as a biological filter, using appropriate media, they are excellent aerobic filters, but also do not allow any anaerobic reduction of nitrates. They are fairly cumbersome to clean, requiring in-line shut-off valves, but in my experience still spilling some water every time they are uncoupled. If they are used as bio-filters they also require a pre-filter to prevent waste from accumulating in them. They do provide good circulation in the tank, and allow UV filters, etc. to be connected in-line. Due to the high organic load (detritus) normally found in these filters, and the small volume of the canister, they lose oxygen very rapidly when switched off. Aerobic bacteria can thus only survive for a few minutes if the power goes out.
These are "new generation" aerobic filters. They are the most efficient aerobic filters available today. They do not build up any waste, and can maintain a very high aerobic bacterial population, without any maintenance or regular cleaning. They do not have any nitrate reducing capabilities, though, as they have no anoxic regions. They also produce nitrates at a very high rate, causing the nitrate level in the tank to increase to a dangerous level sooner than most other filters. Due to the high bacterial population they also have a good reserve capacity, and can prevent ammonia spikes caused by dead organisms in the tank. Unfortunately they cannot survive a "stoppage" for more than a few minutes before the bacteria start to die off. This type of filter has a restricted flow rate (as the "fluidised" medium must remain in the filter, and too fast a flow will wash the medium into the tank (or sump), destroying the filter) and does not contribute much to tank circulation.
These are the filters most widely used today in "Fish Only" marine aquaria. They are very good aerobic filters, usually having enough "reserve capacity" to safely neutralise the decomposition of a dead creature in the tank, without causing a disastrous ammonia "spike". They also have the longest "switched off" survivability, as the media is exposed to air, and must only remain damp for the majority of bacteria to survive. They do tend to accumulate detritus over a long period, and require regular cleaning (every 4 - 6 months) to perform optimally. They do not have any anoxic regions, and does not support nitrate reducing anoxic bacteria. Because they are such efficient ammonia / nitrite reducing filters they tend to produce large quantities of nitrate, causing the nitrate level of the tank to increase over time - in fact many people refer to them as "nitrate factories". The water is usually returned via a dedicated pump, providing some circulation to the tank.
Also known as Foam Fractionators, these "filters" remove DOC from the water before it can decompose, thus preventing the formation of ammonia, nitrite and/or nitrate, to a large extent. I won't go into the principle of their operation, suffice to say that good quality skimmers do work very well. They can also "export" phosphate if used with "Kalkwasser" (a calcium additive). In my opinion a good quality skimmer is a MUST for any system, as it reduces the workload of any other filter.
Depending on rock, sand or both are used, this method is known as the "Berlin", "Jaubert", "NNR (natural nitrate reduction)" or "Modified Berlin" system. This filtration system makes use of efficient skimming (to remove most DOC before it decomposes), so-called Live Rock (for aerobic and anaerobic biological filtration), high water circulation using some power heads, and a very light bio-load. It is mostly used in "Reef-type" tanks, where exceptional water quality must be maintained, with the ideal being an undetectable nitrate level. The "Live Rock" is usually very porous calcitic or sandstone rock. The high water circulation causes a highly oxygenated area in the outer region of the rock, allowing ammonia and nitrite reduction. Deeper into the rock the anoxic condition is ideal for anaerobic bacteria, and nitrate to nitrogen reduction occurs here. The proximity of the two regions also allows for a very efficient nitrate reduction.
The "Live Sand" method works on the same principle, with the sand close to the surface being aerobic, and the deeper region being anoxic. Both types, but especially the "Live Sand", also make use of small organisms, such as worms and Copepods, to eat the (primary) detritus of the fish, thereby also reducing their pollution effect. The "Modified Berlin" system uses both rock and sand, combining their advantages.
Algae consume nitrate and phosphate as food. A good growth of macro algae, or coralline algae, will thus use up some of the available nitrate. If the macro algae is regularly "harvested", the nitrate is removed from the system. Although some aquarists have reported success with the use of "algae filters", separate tanks with algae grown exclusively to extract nutrients, much greater success is usually achieved with the use of a Refugium. I run a section of my sump as a refugium. I have 4 "economy" household power compacts, totalling 80W (the equivalent of 480W of incandescent light), on a reverse daylight photo period.
Others have used "Mangrove filters", using Mangrove plants growing out of the water in the sump, to achieve the same goal. Personally I don't have any experience with these types of filtration, and can thus not comment on their effectiveness.
These are not actually filters. They use ultra violet rays to kill (sterilise) any living organism in the water flowing passed the UV tube. This kills alga spores, bacteria and small parasitic organisms. It also kills the spores of coralline algae, beneficial bacteria and beneficial organisms (potential invert food). It is therefore more applicable in a densely populated fish only set-up, to reduce illness.