The filter you are referring to is called an undergravel filter, or ugf. Water is drawn down through the gravel where the wastes are broken down by nitrifying bacteria. This purifies the water and turns waste into fertiliser. You need a separate pump to run this filter. It is probably the cheapest option when it comes to filtration, along with box filters. These are canisters filled with filter wool etc.
Next would be the internal filters which have a built-in pump, and then the external canister filters.
Your choice of filter often depends on what sort of fish you have and how big your tank is.
Snails are often used to clean up excess waste in a tank but they tend to poo more than they eat and can quickly overrun a tank. Quite a few fish eat snails and they will not last in a tank with pH below 7.0 as the acidic conditions eat away their shell.
I have never used bivalves or mussels in a tank so can't comment on that but if they die they can pollute a tank very quickly.
Do you have a cold or tropical tank? What size is it and what fish do you have?