Here's a link to the kind of sponge filter that I get from Amazon:
Sponge filters work by housing bacteria storing debris and in the sponge. They draw water through the sponges and spit out the clean water. You do not need to add carbon to them.
You will, however, need airline tubing (standard size, this is available at any aquatic pet store or online) and an air pump. I recommend the tetra whisper air pump. You can select a size appropriate to your tank. These air pumps are generally pretty quiet; if you notice that yours is making a lot of noise, there is a way to fix it (some things inside are just misaligned), but you're generally better off exchanging it for a different one. The air pump should be placed somewhere above the water line of the tank in order to prevent backflow (I have mine on a shelf and put a paperback book underneath it to reduce vibrations).
Before setting everything up, I would recommend plugging in the air pump. This let's you check for noise and see if it's actually working (you should be able to feel the air output if you pass your hand in front of it). Unplug it (to shut it off) before setting up the whole system.
To set everything up, you will hook one end of the airline tubing to the filter and the other end up to the pump. It will be a tight fit, as it's supposed to be. I usually have to kind of wiggle the tubing on to both things. You will want to attach the filter to the inside wall of the tank before plugging in the air pump. Then you plug it in and should be good to go.
It's a very simple system, but still requires some maintenance. Every week, when I do a water change, I take out about 30% of my tank water and put it in a bucket. I then rinse off all of my non-porous tank accessories inside the bucket (hides, plastic part of the filter, etc.). After that, I gently squeeze the moss balls a few times in that water to clean them out, and then do the same thing for the sponges.
These are by far the dirtiest thing, so you want to do them last. This helps get rid of the debris the sponges sucked up, but the good bacteria will still mostly stay. You want to get rid of as much debris as you can or the sponges will not be able to adequately draw water through them. You should replace the sponges every 3 months. I personally replace the plastic parts every 6 months or so as well.
I like having the double sponge because you can replace one sponge at a time and leave the other in there so that the tank doesn't crash from losing all of its good bacteria. I often replace one, wait about a month, and then replace the other. To help remember which is which, I usually shove the older one a little farther onto the plastic holder (though be careful that all the holes are still covered up by sponge) so that it creates a noticeable dent in the sponge, even after cleaning.
Hope this helped, and please let me know if you have any other questions!