'Box filter' is a term normally used to describe a form of small air-driven filter designed for small aquariums.
A box filter typically consists of a small enclosure containing a mechanical medium, normally polymer wool based, and a chemical medium such as activated carbon. The activated carbon, or other chemical media, will remove the waste products in the aquarium such as ammonia and nitrite to a limited degree, and will require regular replacement to remain effective.
The filter is attached to an airline and air pump, which, through the movement of air through the filter, creates a lift that draws water through the filter.
Box filters may be useful as emergency filters, temporary filters (e.g. during quarantine or treatment isolation), or for aquariums used for raising fry (the reduced flow ensures the young are not sucked away) but are not suited for long term use as they have little bacteria holding capacity and thus do not provide a long-term stable environment.