Ambassador is an unofficial position created by eRepublik users.
Ambassador's or diplomat's job is to become a representative or delegate of their country in others and are often the gateway that a country uses to communicate with the ambassador's country. They often are very adaptable as they must immerse themselves in the country's society, culture and governmental structure.
Party ambassadors work the same way but their job is to represent their own party in a foreign country's party that shares similar interests.
Neither of these ambassadors have anything to do with the Ambassador Program achievement.