Shah (in Persian: شاه), from the Old Persian word khshathra "king", is the Persian term for a monarch and used by the former rulers of Persia (present-day Iran) as well as the rulers of the Persian Empire. The title roughly translates as Emperor in English as the monarch of Persia was technically the Emperor of the Persian Empire (later the Empire of Iran as Iran was officially known until 1979).
The last Shah of Iran, Mohammed Reza Pahlavi officially adopted the title شاهنشاه Shâhanshâh "Emperor" (literally "King of Kings") during his coronation. He also styled his wife شاهبانو Shahbânu "Empress" - previously, the ruler's consort was known as ملکه Maleke "Queen", (a borrowing from the Arabic term Malika, meaning Queen). The title Shâhanshâh was also used by the Ottomans for a great hero or Emperor.
The English words check, checkmate (with the semitic root "mata" - to die) and chess are all derived from Shah.
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Shah coincidently is also a common last name among people from India, especially the state of Gujarat and the city of Bombay, It usually indicates the person or previous generations were involved in trade or money lending, and is unrelated to the above meaning of King.