"Recognition" is a legal term meaning that the United States recognizes a government-to-government relationship with a tribe and that a tribe exists politically in a "domestic dependent nation status." A federally recognized tribe is one that was in existence, or evolved as a successor to a tribe at the time of original contact with non-Indians.
Federally recognized tribes posses certain inherent rights of self-government and entitlement to certain federal benefits, services, and protections because of the special trust relationship.
Tribes have the inherent right to operate under their own governmental systems. Many have adopted constitutions, while others operate under Articles of Association or other bodies of law, and some still have traditional systems of government. The chief executive of a tribe is generally called the tribal chairperson, but may also be called principal chief, governor, or president. The chief executive usually presides over what is typically called the tribal council. The tribal council performs the legislative function for the tribe, although some tribes require a referendum of the membership to enact laws.
Content Source: The Department of the Interior, Office of American Indian Trust