The Abbasids rose in the year 750 to replace the Umayyad dynasty, by uniting several groups which had grievances against the Umayyads. This included the Shia, who liked the fact that the Abbasids, unlike the Umayyads, had a direct familial tie to Mohammed, many religious figures who thought the Umayyad dynasty had become corrupt and impious, and non-Arabs who resented the domination of major offices in the Caliphate by Arabs.
The dynasty lasted 500 years until it was finally wiped out by a grandson of Genghis Khan, but it's peak had ended at least 300 years before that. Long before the Abbasid Caliphate was officially destroyed, it had ceased to have real power.
There were many causes of the fall, but probably the main one was a failure to find a good solution to the difficult problem of holding together a very large empire in the fairly primitive circumstances of the early middle ages. The solution the Abbasids used was to delegate power to local rulers who passed their own authority on to their sons. These local rulers tended to become independent powers, who gave only formal allegiance to the Caliphate.
