If you haven't already figured it out, terms in the hobby rarely have a solid concrete meaning, and a "Berlin Filtration System" is another one of those things. Basically, it is any method of filtration that utilizes a protein skimmer, liverock and should have a means for mechanical filtration as well. The system utilizes a sump, which is a tank separate from the display that contains all the equipment out of view.
Typically, the sump is located below the tank. Water from the display tank drains down through an overflow or other hole drilled into the tank, (although some people still use siphons for this - not recommended), and is aimed to pour into filter socks, or is poured over a filter pad. Utilizing gravity in this way makes mechanical filtration simple, but some of the heavier particles will remain in the tank as they sink below the drains. Stirring up these particles in your display tank by blasting your rocks with a turkey baster will help the filter remove them in between water changes.
Mechanical Filtration in your typical Berlin Filtration System -
Filter socks are usually made from a polyester blend and allow water to pass through them, but remove any particles over a certain micron size. Usually "100 micron socks" are sold in the hobby, these socks in theory will catch all particles over 100 microns, but in reality the longer they are used the less effective they will be as rushing water opens them up, and particles are broken down to sizes that will pass through the sock. Because of this, they should be taken out and cleaned each week, and depending on use and the quality of the sock, replaced when needed. You should clean the filter socks before water begins rushing over the top of them, if this happens, the sock is completely clogged, and will greatly reduce its effectiveness. To clean a filter sock, turn it inside out and spray it with a hose from the "new inside" to dislodge the particles now exposed on the outside of the inverted sock. Move back and forth, from top to bottom, before rotating the sock, continue until it is almost 100% clean. You can turn it back to its normal shape and begin using it again after it is cleaned, the hose water is very unlikely to cause any problems to your aquarium, it is too small of an amount. There are other ways of cleaning the sock, like letting them sit out in a bucket of strongly chlorinated water overnight, but I think spraying is effective for the first couple of cleanings, and then they should be replaced anyway.
Filter Pads serve the same purpose as filter socks, and are also made of a polyester blend. Filter pads contain a network of material that catches particles as water passes through the mesh. For a filter pad to work effectively, it should be situated in a way to prevent water from going around the mesh. Usually this can be solved by placing the filter pad parallel with the floor, and allowing water to fall over the pad and then down into the sump. As long as the pad is wide and clean enough to handle the water volume falling over it, gravity will cause the water to go through the pad. Vertically aligned solutions, like the filter pad setups in hang on the back filters, care has to be taken to make sure water cannot go around the pad. In time the pad will become clogged with waste, and water will begin covering a greater area before it fall into the sump. When this happens, it is time to clean your filter pad. I like to slide a garbage bag underneath the pad before taking it outside to clean, so that a minimum amount of waste falls into the tank as the mesh is squeezed/manipulated. Arranging your filter pad for easy access in the first place isn't a bad idea.
Both filter pads and micron socks do a good job of removing larger particles from your water, and protecting your pump from hard objects that may damage it. If you have a shortage of footprint area in your sump, micron socks are probably a better fit for you as they utilize both vertical and horizontal space as compared to filter pads that can only utilize one or the other. (Therefore you can pass more water through a finer particle trap than through a filter sock compared to a pad for the same footprint area; all other things equal).
Whether you use a filter pad or a filter sock for mechanical filtration, make sure to always turn off the return pump from the sump before cleaning them to prevent making a mess in your display tank should you make a mistake.
After water passes through the mechanical filtration area and is free of large particulates, it will usually enter the sump and be returned to the main tank, or sucked up by the protein skimmer. However, the aquarist can employ many other means of filtration in the sump, some of which are discussed here.
Protein Skimmers, the Main component of the Berlin Filtration System -
Protein skimming can get really complicated if you want to get into it, so we will stick to the basics. Protein skimmers remove organic compounds, (Dissolved Organic Compounds -"DOCs" to be more exact), that were too small to be trapped by mechanical filtration, but not have been fully broken down. They do this by utilizing a vortex chamber, that places the water in contact with air bubbles that separate hydrophobic particles from the water, such as lipids, fats and other "proteins". (You will here the term skimming "proteins" from the water often in the hobby, even though technically your skimming more than just that). The separated particles rise in the chamber as cleaner water exits back into the sump. Eventually this film of DOCs forms into what is called skimmate, which bubbles over into the collection cup at the top of the protein skimmer. The cup should be cleaned, (with the pump that powers the skimmer turned off), weekly when the filter pads are checked, or sooner if they become full.
After water is skimmed, in the traditional Berlin Filtration System it is returned to the display tank where bacteria on and in liverock consumes ammonia, nitrite and nitrate that are a result of decay of wastes missed by the filter or caused by the tank's bioload. However, most aquarists utilize additional filtration methods to get the most out of the sump area. Some of the more popular methods involve using water polishers, remote deep sand beds, bio-ball and/or bio-wheel areas, canister filters and refugiums. The filtration methods supplement the Berlin filtration system, and help to provide an even more balanced environment for your marine inhabitants. With filtration, the more the merrier.