Right, so the Abbasids were from the Abbasi, or Abbasi family, which hailed from one of the eastern and therefore more Persian provinces of the Islamic Empire. The Abbasids took over in 750 and no one could fully defeat them until 1258 when they were conquered by, wait for it, the Mongols. The Abbasids kept the idea of a hereditary monarchy, but they moved the capital of the empire to Baghdad and they were much more welcoming of non-Arab Muslims into positions of power. And under the Abbasids, the Dar al-Islam took on a distinctly Persian cast that it never really lost. The caliph now styled himself as a king of kings, just like the Achaemenids had, and pretty soon the caliph's rule was a lot more indirect, just like the original Persians. This meant that his control was much weaker, and by about 1000 CE, the Islamic Caliphate, which looks so incredibly impressive on a map, had really descended into a series of smaller kingdoms, each paying lip service to the caliph in Baghdad. This was partly because the Islamic Empire relied more and more on soldiers from the frontier, in this case the Turks, and also slaves pressed into military service in order to be the backbone of their army, a strategy that has been tried over and over again and has worked exactly zero times, which you should remember if you ever become an emperor.