Most people would say that the permanent end to the Abbasid dynasty came in 1258 when the Mongols sacked Baghdad and killed the caliph. However, some could argue that the caliphate continued to exist because the Abbasids relocated to Cairo and continued to claim spiritual authority until the Ottoman Empire conquered Egypt in the 1500s.
If we are going to say that the caliphate ended when the Abbasids lost their actual control over the Muslim world, then we should say that the sack of Baghdad was the event that marked the permanent end of the dynasty. The sack of Baghdad occurred during a time when Hulagu Khan ruled the Mongol Empire. He was the grandson of Genghis Khan and the brother of Kublai Khan. Hulagu increased the size of the Mongol Empire by taking lands to the southwest. When he reached the Abbasid Empire, he offered to allow the caliphs to continue to hold spiritual power and symbolic authority even though the empire would actually be ruled by the Mongols. The caliph ignored the offer, so Hulagu besieged Baghdad and then destroyed the city, killing huge numbers of residents, including the caliph. This was the end of real power for the Abbasid dynasty.