I had always assumed that Air Force One was the name of that specially-built plane with the presidential seal on the tail you always see on TV whenever the President flies somewhere. Not so, apparently, as a recentSlashdot thread has taught me:
Actually, Air Force One is the call sign only if the aircraft is an Air Force aircraft. Other potential names include Navy One (for example, when Bush landed on an aircraft carrier), Marine One, and Army One (for obvious reasons). If not flown by the armed services, the call sign would be Executive One. And yes, when not flying the president or the vice president (Air Force Two in that case), the planes would be referred to simply by tail numbers.
All of which can lead to some pretty complicated situations:
When Nixon left office, the plane he took off on was Air Force One. When he landed, the plane had the tail number call sign. The call sign changed when Ford was sworn in. The White House made special accommodations with the FAA ahead of time to change the call sign mid-flight.