A special emissary or special envoy was a diplomat with a very specific mission. The duties and responsibilities that went with the title were ill-defined so as to allow situational application.
The half-Klingon K'Ehleyr was once granted the title while inside Federation space. Starfleet honored the moniker, because they viewed her as vital to the success of a mission regarding Klingons who still believed the Klingon-Federation War was raging.
Though defining the title was difficult, the power relationship between K'Ehleyr and Jean-Luc Picard was one of rough equality. He was obliged to put USS Enterprise-D at her disposal, and to let her lead the response to the situation. Nevertheless, he still had a say in the precise manner in which the mission was carried out. In particular, she had initially thought the mission would require Enterprise to simply kill the under-informed Klingons, but he successfully insisted on a change of tack. (TNG: "The Emissary")