Skimming, skimming, skimming. Successful tanks can be kept without the use of a protein skimmer (foam fractionator), although for the beginning aquarist I highly advise the use of a good quality skimmer. Skimmers work by forcing air and water to mix causing the organics in the water to stick together. This creates foam which is then pushed out of the skimmer and removed from the system.
Other forms of filtration do work as well. Hang on filters work fine for small tanks if water changes are done on a regular basis. The use of activated carbon is highly suggested as well as it works to remove organics much like a skimmer.
Undergravel and canister filters are not suggested for use. They are a risk to the animals in the tanks due to the fact that anaerobic pockets can develop and release highly acidified water to the system (not to mention stink A LOT!). This is especially an issue when the power goes out for a couple of hours. They also reduce the efficiency of the sandbed to reduce nitrate to nitrogen gas to be released into the atmosphere. Biomedia (bioballs, filter floss, etc) is also generally not used any more. These will cause a great deal of aerobic degredation of ammonia to nitrate, but the nitrate is the end product which causes it to build up in the system to levels that can cause stress to corals.
A good deep sand bed combined with a good quantity of live rock and good circulation provides a great deal of filtration, especially in combination with a good skimmer