The "Berlin method" is a methodology for keeping saltwater reef aquaria. It's named for the hometown of one Dietrich Stuber, who popularized the method by demonstrating that he was successfully keeping living hermatypic stony corals (acropora sp.) - a type of sea life believed impossible to keep in captivity. The Berlin method works so well because it uses the same processes at work on a natural reef to keep the system healthy, rather than relying on more artificial means.
In the bad old days (that is, from the dawn of modern reef keeping to the late 1980s) saltwater aquaria were basically variations on your standard goldfish bowl. Dead rock or bleached coral skeletons provided the backdrop for a few sickly zoanthids, large-polyped soft (LPS) corals, and perhaps a slowly fading anemone, while colorful reef fish swam between them. Bacteria that processed ammonia and other biological waste products lived on bioballs or in wet-dry or trickle filters, which also provided physical filtration. Still, water was thick with nitrate, organic waste, and other things rarely seen on a natural reef; these contaminants were kept in check only by large and frequent water changes. Diseases, brought on by unhealthy conditions and stress, were common, and were treated with copper-based medications. Sensitive organisms, like the aforementioned soft-polyped stony (SPS) corals, or crustaceans, for which the copper medications were lethal poisons, quickly perished in this type of system, while the rest were condemned to a lingering death. This is not to say that aquarists didn't make heroic efforts to keep their animals alive, but their methodology was all wrong - the sea is a vastly different (and in many ways more complex) environment from a freshwater area, and requires a different approach.
The water around a natural coral reef is very nutrient-poor. This explains the amazing diversity of the reef environment - as in the tropical rain forest, the extreme scarcity of resources has produced an environment that is both precariously balanced and host to a staggering number of different species. Clearly, natural sea water is not kept clean of nitrates and disolved organic compounds by a vast array of trickle filters. Instead, the reef is home to a number of life forms that sweep the water clean on their own, with the balance handled by the mechanics of the ocean itself. The best and easiest way to replicate the reef environment in a glass tank is to replicate these factors as well.
Enter the Berlin method, which exchanges mechanical filters and things not found on a real reef (bleached coral, tufa rock, or dolomite gravel) for three things:
A protein skimmer, which uses the unique properties of seawater to export organic compounds before they can decay and enter the biological system,
Live rock, either harvested or aquacultured, which provides both the aerobic and anaerobic environments necessary for the bacteria of a complete nitrogen cycle, and
Aragonite substrate, which mostly looks nice, although it can help maintain the aquarium's water chemistry. As of late, deep sand beds are becoming more popular, as they provide additional biodiversity and denitrification.
That's it. The skimmer pulls organic stuff out of the water before it can rot and poison anything, and the ecosystem in the tank takes care of the rest. It's simple, but simple methods are usually the best. This is not to say that keeping a saltwater tank is easy, of course - but it has become a matter of finesse (balancing salinity, trace elements, and nutrient inputs) rather than manual cleaning and water changes. Best of all, the glass box full of poisoned water and an incomplete, dying ecosystem is becoming a thing of the past. A modern reef aquarium is full of healthy, varied life forms interacting much as they would in the wild. And isn't that the point of having the thing, anyway?