A side firebox is the main identifying component of an offset smoker. Adding one to a regular charcoal grill of the appropriate shape will turn your grill into a DELICIOUS MEAT MACHINE. Typically, the firebox is a chamber shaped like a smaller version of the main chamber and made of the same metal. It's attached at the opposite end of the chamber from the smokestack, where the damper would normally be in an unaugmented cooker. The firebox exists so that it's easier to provide indirect heat and smoke to the main chamber and thus cook your food lower and slower. A side firebox also extends your usable cooking space; it moves the fire out from under your main cooking grates, and could even function as an extra charcoal grill if you need to make burgers for a football team.
Various models of barrel-type smokers that are available commercially can be augmented with a side firebox, sometimes marketed as "for Texas-style smokin'." These may be constructed out of anything from thin sheet metal to 1/4" thick cast iron. Barbecue aficionados will go so far as to construct their own pits out of discarded propane or heating oil tanks, or sometimes food grade steel barrels, old refrigerators, airplane parts, car bodies, and so on. These are the pits you will see built into trailers and used for BBQ teams and catering.