Heres what I remember about filters:
Sand - by far the easiest and least costly. You may only have to change the sand every other year depending on pool conditions. However if you do have an algae problem the sand filter is the worst for cleaning it up. The sand filter catches the largest particles and lets the smaller ones slip by.
Earth a.k.a. - DE - The most efficient filter at catching fine particulates. The is the filtering system if you want the cleanest water. Also seems to be the most work and fraught with the most problems. The bonus to the system is that you change the filter medium when you backwash the filter. You start "almost fresh". This filter will clean up a pool the fastest. Most amount of maintenace. More costly to purchase and most costly to maintain.
Cartridge - many new advances in this area in the past years. The newer cartridge filters have better elements than ones in the past. Some systems can even use earth with the cartridge. Will pick up most particles but not as small as earth filters. More expensive to buy and maintain, the filter elements can get expensive although the new ones are a lot more durable than the older paper elements. Not too much down time if you have a spare element, as you clean one you put another in to run.
The biggest determining factor in whether or not you will have clear clean pool are water chemistry, proper sanitation, and proper filtration. If you have a choice buy a filter one size bigger than you need, it will work better and longer before cleanings are needed. The more that you run the filter the less chance for algae growth. Running the filter 24/7 is not unheard of when the weather is really hot around here.