Unless there is a unit callsign (which must be approved by the FAA, much like airline callsigns), the aircraft will use the service (Navy, Army, Marine, Coast Guard, Guard for National Guard), followed by the tail number. E.g.- Guard 26599. VIP aircraft of all services are given the callsign PAT (for Priority Air Transport). You will here everything from Gulfstreams and Blackhawks carrying a general to a C-5 carrying the Presidents motorcade using this.
Some times we will add "copter" to help distinguish the type for ATC- "Guard Copter 26599".
Most of the FAA approved callsigns have some history to them, or a tie to the unit. Gator is a unit with a tie to U of F; Red Dog a Marine unit with ties to U of G; etc.
Then there are the well known military callsigns such as Air Force 1, Marine 1, Air Force 2, etc.
I think there are some others, but I don't have the General Planning guide that covers all of them with me at the moment.