Each of the filtration types have their place in life. I will try to hit all the pros and cons of each type of filtration system. Each system has its way of dealing with the Nitrogen cycle.

Under Gravel - The undergravel filter is by far considered a beginners filter. It is seen in 90% of all freshwater setups, and in most beginning saltwater fish only systems. This filter simply uses a piece of plastic resting about 1/4 of an inch above the bottom of the tank. This plastic is then covered with rock. There are tubes extruding from the plastic. These are called uplift tubes. These tubes use air bubbles, or powerheads to pull water from the main tank down through the gravel, under the the undergravel plate, then up through the tube and out to the main tank again.
Wet Dry - This filtration is fine for fish only tanks, but it has no place in the reef aquarium Wet/Dry - Back in the "old days" of reef keeping, This method of filtration was common place. There are few respected authors who whole heatedly believe that a wet/dry is of the utmost importance even today. As they do a very nice job at breaking down the organics in a system, the can produce high amounts of Nitrate. And as we all know, this is a no no in a reef system. Some will probably argue with me on the latter statement, but that has been my experience.

Berlin System - This system consists of nothing more than a Protein Skimmer and live rock. This has been the common practice over the last 5 - 7 years. It is also my preferred method of filtration. Live rock should be in the area of 1 LB per gallon for a very porous rock such as Indonesian, and as high as 2.5 - 3lb's for rock not so porous. Such as Floridian. The Berlin system does need to have chemical and mechanical filtration in conjunction with it, but this should be done across all forms of filtration.

Jaubert Method - This system consists of the Berlin system with one added difference. The tank has about a 2 to 3 inch dead space at the bottom of the tank. This dead space is called a "Plenum". The plenum is where the anaerobic bacteria live. They are beneficial to the reduction of Nitrate. Hence this is what is known as a NNR system. (NNR = Natural Nitrate Reduction) During the inception of the Jaubert system, there were many good things to say about it. It seemed to work just fine and did its job of reducing nitrate. However, I have recently, heard of tanks that have been established for nearly 1.5 years, that have suddenly produced a hair algae bloom. The common denominator between all these systems is that they all ran the Jaubert system. This makes ME ( Don't let my opinion sway your decision. There are people out there that can give you more information than I can) somewhat skeptical of using this method. ( personally, I feel that you can get the same results with enough good quality rock. In all the years in the hobby, I have never had a reef tank with any detectable amounts of Nitrate while using a bare bottomed tank.)

Natural System -  This system tries to get us to revert back to the basics so to speak.  It is a matter of only a Live Sand Bed (LSB) or Deep Sand Bed(DSB), live rock, good circulation, and critter maintenance.  The goal with this type of setup is to create a self sustaining ecosystem.
Skimmers, Canister filters, and various other toys are excluded from the list of requirements.  But keep in mind, this is minimalist in nature.  That means you CANNOT have allot of fish with this system, and water changes are a MUST!

Mangroves - Please be a contributor to this site. If you have experience with this method, please write a small article and send it in. You will have your name listed as the contributor at the beginning of the article, as well as have your name listed in the contributor section.