To pick an aquarium filter, there are a lot of different types of filters on the market, you want to get one that best suits the type of tank that you have. The good thing for consumers nowadays is the increased options for turn key aquariums. There are a ton of aquariums out there that the filtration is already built in, so you don't have to consider what type of filter you want to go with. But if you're going with maybe a larger aquarium, something 55 gallons and up, most of those do not come with filters.
When deciding on what kind of filter you want, you want to look at what type of fish you're keeping - whether it's fresh water or salt water. A salt water fish tank filter would typically be a wet dry filter that will go below the aquarium. If it's a reef aquarium it would be a reef sump with a strong mechanical filter, a protein skimmer, a secondary mechanical filter right before the return pump, and some good carbon and a protein skimmer.
For fresh water aquariums there are hang on filters, canister filters, and, again, wet dry filters. There's also under gravel filters which are the least expensive and, in my opinion, work really well and are underrated.
Our aquarium industry kind of came to be where it is because of under gravel filters. It's utilizing your entire gravel bed as a biological and mechanical filter. I can't think of a more efficient use of your gravel bed. The down side is you have these big, ugly pipes all over the back of the aquarium. And you have bubbles going up the lift tubes or big power heads sitting in the aquarium. So there is a slight visual drawback to going with an under gravel filter, but they do work really, really well.
Budget is another concern when going with a filter. A typical canister filter will set you back a couple 100 dollars. Where a hang on filter will do almost as good of a job and be a fraction of the cost at maybe 40 to 70 dollars. But, it's big, it's clunky, it hangs in the back of the tank versus going neatly in the cabinet below the aquarium and only seeing two hoses that run up the back of the tank.
So there are a lot of options which are good, but in the end you probably want to go to your fish store. Look at the options, touch, feel them. See what works with your price range. And also, it's good to see the filter working so you can see the noise level and just kind of get a good feel of if it's something that you'll be able to service. Because if you have to hire somebody to service your aquarium, be prepared to spend 75 to a 150 dollars per cleaning every 2 to 4 weeks. So, make sure you get a filtration that you're comfortable with if you're not so handy.