The Padishah Emperor was the title of the hereditary rulers of the Imperium and the Known Universe, coming from the ancient Persian meaning "Master of Shahs". They were also known as "Emperors of the Known Universe".
The defeat of House Corrino and House Harkonnen at the hands of House Atreides and the Fremen saw Paul Atreides, as the ruler of House Atreides, become the new Emperor in 10193 AG. Theoretically, when Atreides wrested the Imperial Seat from the Corrinos, both Muad'Dib (Paul Atreides) and Leto Atreides II inherited the title of "Padishah Emperor". However, it would appear that neither of them actually used it.
Upon the abdication of Paul Atreides the title of Emperor ceased to exist in its previous sense. The rule of his younger sister, Alia, was always considered a temporary role. And thus she was bestowed the title Imperial Regent until Paul's son Leto Atreides II had come of age.
Upon Alia's death and Leto's ascension, the title of Emperor took on a different meaning. Leto's formidable prescient abilities and slow transformation into a human-Sandworm hybrid saw him acquire the title God-Emperor. After his death, some three and a half thousand standard years later, the title of Emperor became at most a token title, due to the political turmoil that came with The Scattering.
The title ceased to exist after the death of Leto II, when the Imperium was dissolved.