"Abbasid" was the dynastic name generally given to the caliphs of Baghdad, the second of the two great Sunni dynasties of the Muslim empire. The Abbasids defeated the Umayyad caliphs in battle in 758 and flourished for two centuries.
The Abbasids claimed to the be the rightful heirs to the Caliphate based on their descent from Abbas ibn Abd al-Muttalib (566-652), one of the youngest uncles of the prophet Muhammad.
The Abbasid Dynasty ended in 1258, when the Mongol general Hulagu Khan sacked Baghdad, but they continued to claim religious authority from their base in Egypt.