An airbox is a hollow chamber mounted to the intake side of the carburetor, and performs two or three basic tasks: it quiets intake noise which can sometimes be louder than exhaust, it houses and prevents rain from entering the air filter, and on piston-port engines it contains the backwards spitting of fuel when intake ports close. However, the air box adds restriction, which in turn forces the fuel/air mixture to the richer side. This also creates more vacuum across the jet to pull more fuel. It works, but ultimately it can limit top rpm.
Often airboxes are modified or removed completely to improve air flow. If this is done, one must upjet and find a new way of mounting the air filter if necessary. However, you can also redesign an air box to improve flow while preserving it's noise-cancelling effects if you do some Air Box Calculations.