Choosing the right filtration system is one of the most important decisions you will make. In most cases, the filtration system is the key factor that determines the success of your fish tank. This determines the type of livestock, size and quantity that you will be able to keep. This system will also keep the water clean, free from particles like food or waste, and toxins that may harm your precious pets.
In general, there are three different types of filtration that are important in a water system. You need mechanical, chemical, and biological filtration to ensure the success of your tank. During mechanical filtration, particles are removed from the water by forcing it through material that helps to catch all of these particles. The chemical process removes any toxic build up in the water through a chemical resin. Finally, biological filtration occurs when a variety of bacteria convert the byproduct produced by the tank inhabitants into less toxic chemicals. When this process occurs it is known as the nitrogen cycle where the organic waste created in your tank will be converted to ammonia. This is then processed further until it is converted into nitrates that are beneficial to your fish.
To ensure that your tank remains safe for your valuable fish you need to make sure that you have adequate filtration mechanisms in place that will facilitate the different types of filtration. 
There are a wide variety of systems available. However, the most common types are internal, under gravel, power, canister, or wet or dry. Each one of these has their own advantages and disadvantages. At the end of the day, the type of system you choose will depend entirely on the type of aquatic pets that you intend on keeping. Live planted freshwater tanks will most commonly require a mechanical and chemical filtration system. On the other hand, a heavily-stocked African Cichlid tank needs a combination of the three different types of cleaning systems.
Internal systems will often come in a wide variety of styles. They can be placed on the substrate or mounted to the side. As the name indicates under gravel systems are hidden beneath the substrate in your tank. Power filtration systems hang on the back of your tank and will generally use all three varieties of filtering the water. If you use a canister system you only need to place it at the bottom of your tank. Finally, a wet or dry system is placed at the bottom of a tank and requires an overflow system to clear the water.
Before you make a decision about the type of filtration you are going to install into your tank you need to do a little bit of thinking. Look at the type and quantity of livestock you plan on keeping before making a final decision.