Short answers, pulled from the story.
The Abbasid Caliphate rose to power in 750, when the Abbasid Revolution overthrew the Umayyad Caliphate. The first Abbasid caliph was Abu al-Abbas al-Saffah, who descended from Abbas ibn Abd al-Muttalib, one of the youngest uncles of the Prophet Muhammad.
Baghdad was founded in 762 by the caliph al-Mansur on the Tigris River and served as the Abbasid capital for most of their history. It became the center of the Islamic Golden Age, housing the House of Wisdom and over eight hundred doctors by the 9th century, and was renowned across the world as a center of science, philosophy, and learning.
Harun al-Rashid is the caliph traditionally associated with the height of Abbasid power and prestige. During his reign the empire's navy seized Cyprus, his forces won the Battle of Krasos in 806, and he maintained diplomatic ties with the Tang Chinese court. His succession plan, established in 802, ultimately triggered the civil war that first cracked the caliphate's authority.
Hulagu Khan besieged Baghdad beginning on the 29th of January 1258. After the caliph al-Musta'sim surrendered on the 10th of February, the Mongols slaughtered the population and sacked the city starting on the 13th of February. Al-Musta'sim was wrapped in a carpet and trampled to death by horses on the 20th of February 1258, in accordance with a Mongol taboo against spilling royal blood on the ground.
The Abbasid era produced algebra through al-Khwarizmi's Kitab al-Jabr wa-l-Muqabala, the empirical optics research of Ibn al-Haytham's Book of Optics published in 1021, and Avicenna's Canon of Medicine. Abbasid engineers also spread papermaking from China to Europe and developed industrial uses of waterpower, crankshafts, and gears.
A branch of the Abbasid dynasty was reinstated in Cairo in 1261 by the Mamluk ruler Baybars as a ceremonial institution. These Cairo caliphs held no political power but conferred prestige on the Mamluks and were recognized by rulers across the Muslim world until the 15th century. The last Abbasid caliph, al-Mutawakkil III, was taken to Constantinople after the Ottoman conquest of Egypt in 1517.