Air Force One
Formally, Air Force One is an air traffic control callsign that indicates an aircraft that is carrying the President of the United States. In common practice, the name is associated with a United States Air Force VC-25A aircraft, one of two, with tail number either 28000 or 29000. The VC-25 is a highly customized Boeing 747 wide-body transport, with executive working, sleeping, and dining facilities; press and staff seating; and the full range of secure communications needed to support the President in his role in the National Command Authority.
The National Emergency Airborne Command Post (NEACP) is also associated with Presidential missions, but these are E-4B aircraft, also modified Boeing 747s but modified to be flying command posts for the National Command Authority and its military staff.
By the formal definition, as long as the President were aboard, any Air Force aircraft would be Air Force One, just as the executive transport helicopter carrying the President, operated by United States Marine Corps helicopter squadron HMX-1, has the call sign "Marine One". Aircraft belonging to the other services, in this specific role, would be "Navy One" or the equivalent.
An aircraft carrying the Vice President of the United States, without the President, would be "Air Force Two", "Marine Two", or the equivalent. There are no other personal call signs, although senior officials' aircraft might have a "special missions" call sign assigned. It should be noted that certain aircraft carrying senior military officers on a command and control mission may have an unclassified code name, such as LOOKING GLASS for the airborne command post of the Strategic Air Command. That same aircraft usually had a classified, per-mission call sign.
To show the difference between mission and aircraft identifier, when Richard Nixon left the White House for the last time, but before Gerald Ford was sworn in as his successor, Nixon went from the White House to Andrews Air Force Base via helicopter Marine One. He boarded a jet transport from the Presidential unit and took off for his California home, still under the call sign Air Force One. When Ford was sworn in, however, the aircraft reverted to its "tail number" N26000 call sign, but still was designated as a "special missions" flight.