A filter's real job is not clearing the water, it is housing the bacteria that convert ammonia into something harmless. Everything else — polish, flow, mechanical trapping — is secondary to that.
Sponge filters run on an air pump. Cheap, near-silent, enormous biological surface area, no intake to injure fry or shred a betta's fins, and effectively impossible to over-power. They are what we would put on a betta tank, a shrimp tank, a quarantine tank or a fry tank.
Hang-on-back filters are the practical default for most community tanks: easy to service, easy to see, good mechanical filtration. The common mistake is treating the cartridge as disposable — throwing it away throws away your bacteria. Rinse it in tank water and replace it only when it is falling apart.
Canister filters hold the most media and hide under the tank. They suit larger or heavily stocked aquariums, and they are the usual choice for aquascapes because the inlet and outlet can be glass lily pipes rather than a plastic box hanging off the rim.
Match the filter to the fish, not just the volume. Bettas, fancy goldfish and most long-finned fish are poor swimmers and a strong outflow exhausts them. Hillstream loaches and many wild-collected plecos want the opposite — fast, highly oxygenated water. If a filter is too strong for its tank, baffle the outflow rather than starving the tank of filtration.
San Diego tap water is chloraminated, and chloramine kills the bacteria colony living in your media. Rinse media in a bucket of water you just siphoned out of the tank. A filter cleaned in the sink is a filter you have to cycle again.
See also filter media and water treatment.