State Alchemists are alchemists who, once they have passed their tests, are officially sanctioned by the Amestrian Government. They act as a part of the military, respecting the rules and ranks within the command structure.
Alchemists are all given a title upon becoming an official state alchemist, which is given to them by Fuhrer Bradley, and it has to do with whatever field of alchemy they specialize in - for example: one title would include "the Flame Alchemist," which was given to Roy Mustang as a reflection of his talent for his unique abilities.
All State Alchemists, by default, hold the unspoken rank of Major within the Amestrian military once they obtain their title. The symbol of their authority is an official pocket watch that is given to them by Fuhrer Bradley. Many alchemists do rise in the ranks in the military, such as Colonel Roy Mustang and Brigadier General Grand.