I don't have any particular expertise in this topic but one thing that's worth considering in this context is that "the Mongol Empire" was not a monolithic nation-state enterprise, like some sort of medieval Nazi Germany taking over Asia. Their armies operated in large part as a confederation of Mongolic and Turkic steppe peoples and tribes from Mongolia itself to the Iranian plateau to the Russian steppe and the Caucuses, agglomerating those peoples onto themselves as they went along.

Just a quick note: There's an AMA tomorrow in /r/AskHistorians with Jack Weatherford, author of several books on the history of the Mongol Empire -- you may wish to re-ask this question then if you don't end up with an in-depth answer here!

There are three things that should be clear. First, as /u/CptBuck stated, the Mongols were a confederation of Turko-Mongol tribes who lived on the Eurasian steppe. These tribes included, but were not limited to: the Dorben, Barlas, Taichi'ut, Keriates, Tatars, Merkids, Naimans, Oirats, Buriyats, Kiyats, and Tuvans. I'll get to their unification later. Second, the Mongol empire was vast, but land fluctuated. The land borders of the Mongol Empire when Genghis Khan died were not the same as when his sons divided the land and continued his work. Third, Genghis Khan united the various tribes and set out to conquer other tribes and nations. We have to understand that the motivation for his conquests were not necessarily to take over land, but rather to influence people, to seek revenge, and to expand his empire. Allies did exist, but they were largely for political gain. Conquered forces were usually used as 'cannon fodder' for the front lines of Mongol sieges.

Who were the Mongols?

When we talk about the Mongol Empire, we are talking about the umbrella term used to define the various tribes and their affiliates who were united under the rule of Genghis Khan and his descendants. At any point, from the unification of the tribes to the breakdown of each of the successor states, a different definition can be employed to describe them. The Secret History of the Mongols, the main primary source from which most scholars draw their information, tells us that the Mongols (I am generalizing here for the sake of space) originated in the mountain forest near Lake Baikal when a Blue-Grey Wolf and a Red Doe mated. This is the creation myth we have. Their descendants spread out on the steppe and were divided into several ulus or tribes. Genghis Khan's birth name was Temüjin and he was in the Borjigin clan, a subset of the Kiyat. René Grousset writes:

The Mongols proper, in the restricted, historical sense of the word, of whom Jenghiz Khan [this is another way to spell Genghis Khan] was one, made their seasonal migrations in the northerwest of modern Outer Mongolia, between the Onon and Kerulen.

Only once before Genghis Khan united the Mongol tribes had unification been successful. We believe that his father, Yesugei, was descended from the line of the Great Khan who first united the Mongols.

What makes a land empire?

So, the Mongol Empire expanded and contracted as most empires do. I've linked several maps that will show the expansion of the land empire. First, a map that shows the empire at the time of Genghis Khan's death and the borders of the empire under Kublai Khan. Note that the empire under Kublai Khan is not a consecutive one. The khans of the Golden Horde in Russia (Kipchak Khanate) were not directly under the rule of Kublai.

This map shows the empire in 1294. I would tend to agree with this map, though I'm only including it for reference. The Kipchak Khanate was active until 1783 and the Chaghatai khanate arguably lasted until the 17th century. The Ilkahnate, established by Hulegu, fell in 1335. The Yuan dynasty ended in 1368. These numbers may not mean much, but the Mongols (or some descendants of them) held territory in Europe, Central Asia, and China for a number of years after the death of Genghis Khan. All of this is to say that the Mongol Empire was vast. I think that understanding this helps to understand the context of your question.

Motivations for Conquest

In some instances, a feud with another nation or an affront had been committed against Genghis Khan. These events oftentimes triggered reactionary, violent responses from the Khan. A few examples are the fate of the Tatars (they poisoned his father), the Merkids (they kidnapped his first wife, Borte), and the Khawarizm empire (Mongol emissaries and traders were killed by a governor and the Shah laughed it off). There were also political reasons for waging war and conquering land. Example: The conquest of the Jurchen in northern China.

Concerning the Tatars- This is copied from one of my previous answers on the Mongols:

The Secret History of the Mongols tells us that Temüjin was 9 years old when he set out with his father to find a wife. Yesugei (his father) left Temüjin with the family of his bride to be at the request of the father of the bride. Yesugei began to travel back to his yurt and saw a band of feasting Tatars. It was a steppe custom that if someone passes through your camp and you are cooking a meal, you must share your food with the person and they must eat the food you have offered them. Yesugei took the food from the Tatars and fell ill. Three days later he arrived at his yurt and asked someone to take Temüjin back home. He died not long after this request was made.

Naturally, Genghis Khan later sought revenge on the Tatars and effectively wiped them out.

Concerning the Merkids- (Again copied from an earlier answer):

At 16, he decided to go back to his intended wife's family to claim her as his bride. Her name was Bortë. He found Bortë as he had left her and they returned to his camp to be wed. After an undetermined amount of time, Temüjin and his family were sleeping in their yurt when the tribe from which his mother had been kidnapped decided to seek their vengeance. The Merkids attacked in the early morning, but one of the elderly women in Temüjin's camp alerted the family to the danger. She attempted to hide Bortë in a wagon of furs while Temüjin used traditional steppe tactics to draw the enemy away from the women. He rode away from camp, but the attackers stopped the old woman with her cart of furs and kidnapped Bortë. Temüjin spent another undetermined amount of time in the mountains before returning to rescue Bortë from the Merkids. Temüjin drew on a new alliance with Ong Khan, another tribal leader, and launched a raid on the Merkids. The Secret History of the Mongols describes in great detail that during the raid, Bortë was thrown into the turmoil but she heard a voice in the distance. She knew it was Temüjin and they were reunited.

After this episode, the Merkids met a similar fate as the Tatars.

Concerning the Khwarazemid empire- (More about this here)

The Mongols dealt a hefty blow to the Khwarazemian empire in Central Asia after a dispute over envoys that were killed. In short, Genghis Khan destroyed the Kara Khitai and wanted to open trading routes through the Khwarazemian empire. He sent traders and ambassadors to the Shah, but these people did not make it to Samarkand. They were robbed on the road (this was already an offense of the highest nature) and sought help from the governor of Otrar. Iralchiq refused them help, declared them thieves, and killed all of them. Word was received by Genghis Khan and war ensued. The impertinence of the governor of Otrar and the subsequent dismissal of the events by the Shah angered Genghis Khan. I wrote about this event and the war in detail in this thread.

Political reasons also fueled the Mongols want for land. In the early days of Genghis' reign, the Jurchen caused trouble for Genghis' Mongols and the Khitans. The strategy here was to undermine the authority of the Jurchen. They overthrew the Khitans several generations before. By announcing that he aimed to restore the old Khitan royal family, Genghis Khan won the Khitan's support during the subsequent fighting.

Allies

Allies did exist during the course of the Mongol Empire. In the beginning, they were the various tribes that united to make up the Mongol Empire. People like the Khitans threw their weight behind Genghis Khan because they believed he could help to restore peace to their home lands. Some allies were probably not allies. They were forced into service. Oftentimes the conquered forces, instead of being put to death, would choose to surrender and be put on the front lines of Mongol sieges.

Key Takeaways

The Mongols were really a conglomeration of Turkic and Mongol tribes and their land empire was vast. When reading about the Mongols, remember that their empire expanded and contracted, as most do. Motivations for conquering varied from political to personal and everything in between. Allies did exist in the Mongol Empire, but these relationships were tenuous at times.

I rambled a lot and I'm not sure if I answered your question exactly. I'll go ahead and plug Jack Weatherford's AMA as /u/NMW stated. He'll be here tomorrow. You might get a better answer from him. His book does include a lot of information about these events.

Other books you might consult are below.

Suggested Readings

Genghis Khan: Conqueror of the World by Leo De Hartog The Mongols by David Morgan Genghis Khan: His Life and Legacy by Paul Ratchnevsky The Empire of the Steppes: A History of Central Asia by Rene Grousset Genghis Khan and the Making of the Modern World by Jack Weatherford

EDIT: Math and formatting.

Follow up question. It was common for Chinese states to conspire against each other during the Warring States period. Did the Mongols use diplomacy to divide Chinese states? Like did they make any deals with the Song to keep them from helping the Jin?