An air terminal box is typically part of an HVAC duct distribution system and is used to control or modulate the amount of air delivered to downstream ductwork. An air terminal box is often referred to as an "air flow regulator".
A VAV box - pressure independent is an air terminal box that includes a means to reset the volume automatically to a different control point in response to an outside signal such as thermostat: air-flow rate is independent of supply pressure.