Genghis Khan's empire never collapsed. In fact when he died it was still growing. Ogedei, Mongke, and Kublai expanded the original empire which included everything from Mongolia to eastern Europe into China, Tibet, Korea, Vietnam, and Burma. Vast territorial gains even after the death of Genghis. So technically speaking, it doesn't qualify as a collapse like Alexander's empire did, in the way that there was no discernible heir to the succession of the throne.

After the Kublaids were defeated in China by rebellions, the Yuan moved back to Mongolia where they would set up shop and would be known as the Northern Yuan. There they would compete and enter into conflict with various other Mongol powers, the Ming Dynasty, and eventually the Manchus.

The Mongols were roughly split between the western Mongols and the Khalka Mongols who were made of the eastern descendents of the empire. Western Mongols formed a confederation known as the Oirat, which controlled a large portion of central Asia and at times Mongolia as well, directly and indirectly. They had varying successes against the Ming but were never able to successfully launch an invasion like Kublai.

http://en.wikipedia.org/wiki/Tumu_Crisis

After this the Oirat went into decline and was swallowed by a resurgent Yuan Empire headed by Dayan Khan. This would be the last time "Mongolia", as defined by the people who lived there, would become a power to be reckoned with. However like the Oirats, it had lukewarm success against the Ming and never consolidated its victories. Like most Khanates that defined the period, it went into decline soon after Dayan's death.

In the end Mongolia fell into sway of the Manchus and would be ruled by them for another two hundred years before the fall of the Qing Dynasty, which is when modern Mongolia began to show up. Nationalism, patriotism, and independence movements had always existed, but the fall of Qing as well as the arrival of a new backing power in the form of the Soviet Union gave them the push they needed to become a country. Despite this, Mongolia was split, with "Inner Mongolia" going to China, and "Outer Mongolia" becoming the country of Mongolia. This is kind of weird if you get into the details because most Mongols actually live in Inner Mongolia and had deep roots there from as far back as the Yuan Dynasty, outnumbering outer Mongols by something like 5 to 1.