Chemistry determines the optimum relationship between the fuel and the material, among other variables.  The reverberatory furnace can be contrasted on the one hand with the blast furnace, in which fuel and material are mixed in a single chamber, and, on the other hand, with crucible, muffling, or retort furnaces, in which the subject material is isolated from the fuel and all of the products of combustion including gases and flying ash. There are, however, a great many furnace designs, and the terminology of metallurgy has not been very consistently defined, so it is difficult to categorically contradict other views.
The applications of these devices fall into two general categories, metallurgical melting furnaces, and lower temperature processing furnaces typically used for metallic ores and other minerals.
A reverberatory furnace is at a disadvantage from the standpoint of efficiency compared to a blast furnace due to the separation of the burning fuel and the subject material, and it is necessary to effectively utilize both reflected radiant heat and direct contact with the exhaust gases (convection) to maximize heat transfer.  Historically these furnaces have used solid fuel, and bituminous coal has proven to be the best choice.  The brightly visible flames, due to the substantial volatile component, give more radiant heat transfer than anthracite coal or charcoal. 
Contact with the products of combustion, which may add undesirable elements to the subject material, is used to advantage in some processes.  Control of the fuel/air balance can alter the exhaust gas chemistry toward either an oxidizing or a reducing mixture, and thus alter the chemistry of the material being processed.  For example, cast iron can be puddled in an oxidizing atmosphere to convert it to the lower-carbon mild steel or bar iron. The Siemens-Martin oven in open hearth steelmaking is also a reverberatory furnace.
Reverberatory furnaces (in this context, usually called air furnaces) were formerly also used for melting brass, bronze, and pig iron for foundry work. They were also, for the first 75 years of the 20th century, the dominant smelting furnace used in copper production, treating either roasted calcine or raw copper sulfide concentrate. While they have been supplanted in this role, first by flash furnaces and more recently also by the Ausmelt and ISASMELT furnaces, they are very effective at producing slags with low copper losses.