To start with, you may consider entry level units that are affordable and suitable for small sized tanks. Aquarium air pumps which are now mainly used for water aeration can be used to power small sponge based or under gravel filters. These pumps suck water through the sponge or gravel where impurities build up gradually before being manually removed. Not removing the impurities on a regular basis can lead to an increase of toxins in the water which can be harmful to fish and plants, as well as reducing the amount of oxygen in the water.
Powered internal filters are a much more popular choice now for medium sized tanks thanks to their efficiency and low cost. A typical starter option might be an internal canister filter, a popular variety that can be easily fixed to the inside of your tank. These contain a sponge which biologically and mechanically filters water as it is pumped through. Better models will have more than one sponge to filter different impurities. Many internal filters allow you to install your own media, such as carbon filtration, which will support helpful bacteria growth and polyfilters which removes different toxins from the water.
If you have a larger tank it will be necessary to pay a bit more and install an external filter or sump. These can be easier to maintain since they are outside the tank itself; you can discreetly hide them in a cabinet meaning they aren’t in the way and taking up no extra space in your aquarium. External filters are much more powerful, plus they are easier to customise with more media and provide more space for bacteria to grow.
The only thing to be wary with on some filters is that the return flow might be too disruptive to sensitive environments. Delicate corals, and certain small fish like Siamese fighters can suffer if the water flow is too strong. Powerful filters can create a lot of movement and the strong flow can tire or even kill fish like these as they struggle against it.
There are even more options available if you want to add more powerful filtration to the tank. For example, special filters like protein skimmers are ideal for marine aquariums, especially when used with live rock. Protein skimmers remove organic compounds from the tank before they break down and cause water quality issues. Water is pumped through them and impurities are filtered out into a removable section which is easy to empty.
Alternatively, if you have the funds, a more complex filtration system can be set up in your sump with different elements built in depending on your specific requirements.