Background
Hulagu marched out with perhaps the largest Mongol army ever assembled. Among his subsidiary generals was Kitbuqa, a Christian. Hulagu easily destroyed the Lurs, and his reputation so frightened the Assassins (also known as the Hashshashin) that they surrendered their impregnable fortress of Alamut to him without a fight.
Hulagu probably always intended to take Baghdad, which the Mongols had been meaning to attack for over ten years (see Eljigidei), but he used the caliph's refusal to send troops to him as a pretext for conquest, since his brother the Great Khan had ordered him to be merciful to those who submitted. Hulagu sent a message to the caliph, Al-Musta'sim, containing the following (trans. John Woods):
"When I lead my army against Baghdad in anger, whether you hide in heaven or in earth
I will bring you down from the spinning spheres;
I will toss you in the air like a lion.
I will leave no one alive in your realm;
I will burn your city, your land, your self.
If you wish to spare yourself and your venerable family, give heed to my advice with the ear of intelligence. If you do not, you will see what God has willed."

The Assassins and marching on Baghdad

Main article: Battle of Baghdad (1258) Battle of Baghdad
As far as damage done, the sack of Baghdad by the Mongols made the sack of Rome by Alaric look kindly. The Grand Library of Baghdad, containing countless precious historical documents and books on subjects ranging from medicine to astronomy, was destroyed. Survivors said that the waters of the Tigris ran black with ink from the enormous quantities of books flung into the river. Citizens attempted to flee, but were intercepted by Mongol soldiers who raped and killed with abandon.
Although death counts vary widely and cannot be easily substantiated, a number of estimates do exist. Martin Sicker writes that close to 90,000 people may have died (Sicker 2000, p. 111). Other estimates go much higher. Muslim historian Abdullah Wassaf claims the loss of life was several hundred thousand or more. Ian Frazier of The New Yorker estimates of the death toll have ranged from 200,000 to a million.
The Mongols looted and then destroyed. Mosques, palaces, libraries, hospitals — grand buildings that had been the work of generations were burned to the ground. The caliph was captured and forced to watch as his citizens were murdered and his treasury plundered. The caliph was trampled to death. Marco Polo reports that Hulagu starved the caliph to death, but there is no corroborating evidence for that. Most historians believe the Mongol accounts (and Muslim) that the Mongols rolled the caliph up in a rug, and rode their horses over him, as they believed that the earth was offended if touched by royal blood. All of his sons but one were killed. Prior to this, the Mongols destroyed a city only if it had resisted them. Cities that capitulated at the first demand for surrender could usually expect to be spared. Cities that surrendered after a short fight, such as this, normally could expect a sack, but not complete devastation. The utter ferocity of the rape of Baghdad is the worst example of Mongol excess known. (It is said some Chinese cities suffered a similar fate, but this is not documented).
Baghdad was a depopulated, ruined city for several centuries and only gradually recovered something of its former glory. Of all the Mongol Khans, he is, for obvious reasons, the most feared and despised.
Even today, Baghdad residents invoke Hulagu Khan's name as the war and occupation wage on. Several terrorist, militant and insurgent groups refer to U.S. President George W. Bush as a modern day Hulagu Khan.

Sack of Baghdad
Thus was the caliphate destroyed, and Mesopotamia ravaged; it has never again been such a major center of culture and influence. The smaller states in the region hastened to reassure Hulagu of their loyalty, and the Mongols turned to Syria in 1259, conquering the Ayyubids and sending advance patrols as far ahead as Gaza. Egypt's turn seemed next, but the death of Mongke forced Hulagu and most of his army to withdraw. The succession crisis that followed was the most ruinous to date. Indeed, although the succession was finally settled by imprisonment of one of his brothers, and another elevated to Great Khan, (Kublai Khan), the truth is that after 1258 there was no unified Mongol Empire, but four separate kingdoms, including the Il-Khanate of Persia established by Hulagu.
In the meantime, the Mongols led by Kitbuqa had fallen out with the crusaders holding the coast of Palestine, and the Mamluks were able to ally with them, pass through their territory, and destroy the Mongol army at the Battle of Ain Jalut. Palestine and Syria were permanently lost, the border remaining the Tigris for the duration of Hulagu's dynasty.
Hulagu returned to his lands by 1262, after the succession was finally settled with Kublai as the last Great Khan, but instead of being able to avenge his defeats, was drawn into civil war with Batu Khan's brother Berke. Berke Khan, a Muslim convert, had promised retribution in his rage after Hulagu's sack of Baghdad, and allied himself with the Mamluks. Historian Rashid al-Din quoted Berke Khan as sending the following message to Mongke Khan, protesting the attack on Baghdad, (not knowing Mongke had died in China) "he has sacked all the cities of the Muslims, and has brought about the death of the Caliph. With the help of God I will call him to account for so much innocent blood." (see The Mongol Warlords, quoting Rashid al-Din's record of Berke Khan's pronouncement; this quote is also found in The Mamluk-Ilkhanid War). When Hulagu massed his armies to attack the Mamluks and avenge Ain Jalut, Berke Khan initiated a series of raids in force led by Nogai Khan which drew Hulagu north to meet him. Hulagu Khan suffered severe defeat in an attempted invasion north of the Caucasus in 1263. This was the first open war between Mongols, and signaled the end of the unified empire.

Death of Hulagu Khan

Boyle, J.A., (Editor). The Cambridge History of Iran: Volume 5, The Saljuq and Mongol Periods . Cambridge University Press; Reissue edition (January 1, 1968). ISBN 0-521-06936-X. Perhaps the best overview of the history of the il-khanate. Covers politics, economics, religion, culture and the arts and sciences. Also has a section on the Isma'ilis, Hulagu's nemesis.
Encyclopedia Iranica has scholar-reviewed articles on a wide range of Persian subjects, including Hulagu.
Morgan, David. The Mongols. Blackwell Publishers; Reprint edition, April 1990. ISBN 0-631-17563-6. Best for an overview of the wider context of medieval Mongol history and culture.