There are two basic filter types that are used for fountains.
A sand filter uses a bed of uniform size sand to pass the water through from top to bottom and when the sand gets dirty you rinse it out (called backwashing). When you backwash, you flush a certain amount of the water in the fountain away so the volume of the fountain would have an impact on the type of filter you use.
A cartridge filter utilizes a material often paper or polypropylene that filters out dirt and particulate. When you clean it, you don’t have to flush it but you do have to take it out of its housing and wash it off. This takes longer than backwashing and while it still uses some water, it does not use as much as backwashing.