1077's summary is basically my thoughts, so enough on that.
Let me ask a stupid question. What is the purpose of a filter if no carbon is used.
As others have mentioned, filtration can be mechanical, biological and/or chemical. But before getting to those, a side issue is the filter itself, which performs perhaps its most important role in simply moving the water through the tank. This allows the temperature to be more even. It also ensures that oxygen-rich water is carried to the bacteria throughout the tank [more on this below]. In planted tanks it ensures that nutrients (which are in the water) are dispersed to the plants' leaves and roots where they are needed, and it keeps the leaves free of particulate matter that would inhibit respiration. It is quite possible to have a healthy aquarium with no external filtration whatsoever, but this requires a stable biological balance between fish, plants and bacteria and is somewhat outside this immediate question.
Mechanical filtration is simply the passing of water through media to remove suspended particulate matter. All media in the filter perform this task, be it ceramic disks, gravel, rocks, floss, pads, whatever. This plus the afore-mentioned water movement is the only filtration encouraged in planted tanks, since the plants themselves handle the next two subjects.
Biological filtration is generally considered to be the nitrification cycle whereby nitrosomonas bacteria consume ammonia and produce nitrite, and nitrospira bacteria consume nitrite and produce nitrate. Ironically, in most tanks there is more biological filtration occurring outside
the filter itself than inside the filter, since nitrifying bacteria colonize all surfaces covered by water. Each grain of the substrate, be it gravel, sand or whatever, will be colonized by these bacteria, as will every plant leaf, piece of wood, rock, decor and the tank walls themselves. This ties in with the water movement mentioned above; bringing water with oxygen to the bacteria is critical to keep them alive, whether they are in the filter media or throughout the tank.
Nitrosomonas and nitrospira bacteria will only exist at the level necessary to handle the available ammonia and nitrite respectively; when more ammonia/nitrite appears, the bacteria increase by binary division, and if the ammonia/nitrite should decrease, the respective bacteria will die off accordingly. Plants are quicker at grabbing ammonia (as ammonium) and in planted tanks the nitrifying bacteria is considerably less in number than in tanks without plants (all else being equal). Which is why we do not encourage
biological filtration in well-planted aquaria--there is no need for it, and it competes with the plants.
Last we have chemical filtration, which may be defined as altering the water's properties through the use of some chemical substance. Activated carbon, ammonia-absorbing rock material, etc. are all forms of chemical filtration. The idea is to remove something from the water. Any form of chemical filtration is unnecessary in a well-planted aquarium, and in fact is detrimental, since all sorts of good things (nutrients primarily) can also be removed. And, plants perform this task far better than any filter. Plants have the ability to take up--which is not the same as assimilation of nutrients--common toxic substances that may occur in an aquarium, such as heavy metals from the tap water, ammonia produced by the fish and bacteria, etc, and detoxify them something akin to a water conditioner that detoxifies such substances. When plants are present in sufficient quantity, there is absolutely no need for regular chemical filtration, and it should only be employed to remove specifically-added substances such as medications following treatment.