Awwww yis. The question I dream of. I'll try and give an overview of the whole thing and then zero in on Iran here as /u/Jasfss has already done a good job on China.

Ok firstly be aware that the overview I give is a general one. I'm covering and empire that covered half of Eurasia and lasted roughly 150 years (depending on your count).

So how did it work? The Mongols were fundamentally very adoptive. When they rocked up and conquered somewhere they were usually pretty content to leave the basic bureaucratic structure as was. Their main addition was appointing someone in an overseer/governor role to keep an eye on things. These guys were known as shahna, basqaqs or darugachi depending on where they were (although sometimes the terms mean slightly different things this makes research a bit of a nightmare). So who were these basqaqs? In the early stages when the Mongols had basically no experience of governing sedentary populations they made heavy use of educated Uighurs and Khitan. The former was a Turkic tribe with some bureaucratic tradition. The latter group was the remnants of an old Chinese dynasty that had nomadic origins, some of them were still present but a lot had fled to Cenrtral Asia at the end of their dynasty and established the Kara-Khitai kingdom. However as time goes on we see other groups and a fair few Mongols assume powerful positions as well. Overall the system was pretty meritocratic talent had a good deal of scope to rise regardless of background, except in China where /u/Jasfss has discussed the racial policies for government officials.

Now what of the Ilkhanate specifically (an area encompassing Iran, Azerbijan, as well as bits of Iraq,Turkmenistan, Afghanistan and Pakistan)? Well for quite a while its pretty loosely governed, there was a governor appointed by the centre but not much in the way of established Mongol government. Furthermore its appears to have been ruled in a highly exploitative manner. In the reign of the fourth Great Khan, Mongke, a prominent qadi (Islamic judge) comes to the Mongol court in Karakorum and appeals to him to reign in the excesses of the local governor. As Mongke was a pretty major reform, cracking down on corruption in the court and regularising taxes, he agrees. He sends over his brother Hulagu partly to butcher the threatening Hashashin (a bizarre Shia group which assassinated prominent political figures and allegedly consumed huge quantities of hash), continue conquests in the Middle East and bring order to Iran and related areas.

Upon arrival Hulagu pretty much, however local elites retain a substantial decree of autonomy if they swear loyalty to the Mongols. However there is also a centralised bureaucracy these would be drawn mainly from the old Islamic system of recruitment, however we also recruiment of outsiders as well. Rashid al-Din, a very prominent minister, was a Jewish convert to Islam who was originally a physician. Another interesting example is Bolad a prominent Mongol from the Yuan court who came as part of an embassy and ended up as a Ilkhanate courtier.

I can discuss this a little bit from the China side: from the initial invasions through the Yuan Dynasty established by Kublai Khan. The other regions of Mongolian conquest will have to be taken up by individuals far more qualified than myself.

Initially, the Mongolian conquests in the region of China started with the capturing of the lands of the Western Xia (西夏) which were Tanguts and the Jin (金) in the north who were Jurchen; Neither of these empires were ruled by what were considered ethnic Chinese, and most of the ethnic Chinese populations lived in the south under the Southern Song dynasty (there was only one Song dynasty, don't be confused. After first the Liao and then the Jin conquered northern China, the Song had retreated to the south and much of the population followed suit). Eventually, Kublai Khan would conquer the Southern Song and establish the Yuan dynasty.

When the Mongols sacked cities, there was a sort of ultimatum given to the city: Either surrender immediately and pay tribute, or be wiped out. There was no real third option, capture was given as an absolute (this was one of many ways the Mongols used psychological warfare to their advantage). As stories of the conquest spread, more and more smaller cities took up the option to surrender. The Mongols would then often take specialized workers into their ranks (many siege engineers came from the Jin) and recruited troops from the captured populations, either by free will or by threat.

As far as the Yuan Dynasty goes, because of the vast rural agricultural nature of China and the very large population, it was a nightmare for the Khanate to actually rule over the area. The population itself was divided into four major groups: the Mongols, the "Central Asian Auxiliaries" (Naimans, Uighurs, Turkish peoples, and Tanguts), North Chinese, and South Chinese. The Mongols themselves remained the ruling class, while the Auxiliaries took over the roles of government workers, soldiers (as mentioned), and merchants. Additionally, since the Mongolians were often not literate in Chinese, much translation work was taken up by the Uighurs in bilingual government offices. So, while the government structure and reforms were modeled after Chinese dynasties, there was very little Chinese involvement in the government (it was more of a martial rule, as Kublai Khan established the emperor as more of a monarch). Another problem is that the Southern Chinese especially were strictly forbidden from engaging with many aspects of society under Mongolian rule. They were forbidden to marry Mongolians, carry arms, and at certain points forbidden from learning Mongolian or any other foreign language.

Cambridge History of China or Kenneth J. Hammond's From Yao to Mao series covers the Mongol invasions and the Yuan dynasty. I would write a bit more here, but I'd have to cross reference with a book I don't have sitting around right now unfortunately: Mongolian Rule in China: Local Administration in the Yuan Dynasty by Elizabeth Endicott.

The first thing to know is that the mongol empire split into four major "khanates," which were mostly separate after Chinggis Khan died. There was the Yuan dynasty in china, the khanate of the Golden horde (greatest name ever) in Russia, the ilkhanates in Persia, and the khanate of Changtai in the steppes of Central Asia.

The Yuan dynasty functioned similarly to other Chinese dynasties, however, the Mongols stayed separate from the conquered Han peoples. Intermarriage was outlawed, as well as learning the Mongol language. The biggest change was the destruction of the civil examination system, as well as one of the falls of Confucianism. The Yuan dynasty also refused to use the native Chinese in high level positions, using foreigners instead.

In the khanate of the Golden Horde, the Mongols did not interact with Russia very much. They preferred to use the pastures by the Black sea, and usually just forced Russia to pay large tributes.

Rakony is far more qualified to answer about the ilkhanate, all I can add is that the Mongols adapted pretty closely to Persian culture, including occasionally converting to Islam.

Don't forget that the Mongol homeland was defended by the Changtai khanate, as well as being a very harsh and unforgiving place- one of the characteristics that made the Mongols so powerful was their ability to adapt to different conditions, and thrive. There wasn't really a threat of anyone trying to strike back at the steppes.

Finally, the problems with maintaining these lands did exist, and are a large part of the reason the Mongol empire collapsed very quickly- the empire could not be maintained, and only lasted about 200 years.

Source: I used Traditions and Encounters by Bentley and Ziegler.