A ductless range hood is just what the name implies, there is no ducting to the outside of the home. Air that is drawn into the range hood is cleaned, most commonly by charcoal filters.
For many homeowners the inconvenience of having to clean and replace charcoal filters is not as onerous as having to figure out a method of running duct to the outside of the home.
In other cases, the expense of running duct work is outside of the budget.
From an aesthetic standpoint, it is almost impossible to tell whether any specific range hood is being vented to the outside or is ductless, by its external appearance. Many manufacturer's offer the externally identical range hood in both duct and ductless versions, in many cases an individual unit is convertible from one to the other.
Figure 1 shows a range hood that is convertible. If ducts are connected the air flows through the range hood and out through the top where the duct would be mounted. Without ducts the air is directed through the range hood and out through the front of the unit.
Figure 1 - Air flow in a convertible range hood
The best units use a charcoal filter, the economy models use a piece of foam. Foam filters will not remove odors. In either case the filters must be changed on a regular basis. Besides that they will hold odors and eventually disseminate them back into the kitchen, if not replaced, they are also capturing grease that can and will become a fire hazard.
Range hoods that are convertible will generally require the addition of carbon filters and some metalwork. Many manufacturer's sell a conversion kit that converts from a ducted range hood to a range hood that is ductless, as shown in Figure 2.
Figure 2 - Range hood conversion kit
In my opinion the ductless range hood does not perform as well as range hoods that are vented, as charcoal filters become clogged with grease easily and although the range hood is supposed to remove odor the grease laden charcoal filter can be the cause of the odor Ductless range hoods also have the inherent problem that they deliver moisture laden hot air back into the kitchen.
Ductless range hoods are available for all of the common range hood configurations; under cabinet, wall and island styles.
Additional information on kitchen range hoods