In most cases, the receiver is the only ATF-controlled part of the gun as it is marked with the gun's serial number. From a legal standpoint, therefore, the receiver is the gun. In the case of a gun with two receivers, the lower receiver is typically the serialized part. The exception to this rule is in the case of a short barreled gun which is regulated in totality by the ATF.
Generally, the term receiver is used in regard to long guns. The receiver of a handgun is usually called the frame.