Bear in mind that Mongolia has always been a sparsely populated and out-of-the-way part of the world. It briefly obtained great geopolitical significance in the 13th and 14th centuries, but that was a blip on the radar; nobody had "heard much about it" before the Mongol Empire either.

Also, the Mongolia of today isn't the same as the Mongolia of the 13th century. In fact, in Genghis Khan's time there wasn't such a thing of Mongolia. Before he came along the Mongols were one of hundreds of tribes of the eastern Eurasian steppes, and a minor one at that. But it happened to be the tribe that Genghis Khan came from, and so it was the name that was taken by the confederation he built and the empire they conquered. After that "Mongolia" came to refer to the whole region, and for convenience I'll stick to that terminology.

So, in it's heyday Mongolia was the seat of an empire that ruled most of Eurasia. Genghis Khan's immediate successors built a city called Karakorum not far from where their illustrious ancestor was born. Built because of course until then the Mongols and their neighbours were entirely nomadic and there weren't any other cities for thousands of miles. However, given the size of the empire, in practice political power had to be very decentralized. Following nomadic tradition, Genghis Khan had divided his empire into quasi-independent khanates. The khans would return to Karakorum to elect a Great Khan (khagan), whom they were nominally vassals of, but most of the empire was actually governed from the conquered cities where the khans had their courts – Sarai in Russia, Tabriz in Iran and Almaliq in Central Asia. The Great Khan in Karakorum effectively only had direct control over the eastern empire, i.e. Mongolia and China. Never particularly unified to begin with, it's not surprising that the empire quickly began to disintegrate.

The last Great Khan to exert any real control over the other khans was Genghis Khan's grandson Möngke Khan. He was succeeded by his younger brother Kublai Khan, who quickly realised that struggling to bring his cousins in the west into line was a lost cause. Before being elected Great Khan he'd been the governor of the Mongols' holdings in northern China, so rather than continue ruling from a shabby little town way out in the steppe with rapidly fading geopolitical relevance, he relocated his power base there, founding a new capital called Khanbaliq – modern Beijing. He also significantly expanded Mongol rule in China with the conquest of the southern Song dynasty, reuniting the country. To celebrate he proclaimed himself Emperor of a new dynasty, the Yuan dynasty, a title that came to eclipse that of the Great Khan of the Mongol Empire. Mongolia itself was still under his control of course, but it was a province of a Mongol-ruled Chinese Empire rather than the imperial centre.

The fragmented Mongol khanates didn't last very long. The Yuan dynasty was overthrown by a Han Chinese rebellion in 1368 and the surviving Mongol elite fled back to their ancestral homeland. They maintained the pretence of being the Yuan dynasty for a while, but Mongolia was well and truly off the world stage again. The Mongol Empire was the last great steppe empire; the military dominance that steppe nomads had exerted over their sedentary neighbours for centuries was steadily eroded with things like the advent of firearms. Mongolia itself fell back into small regional confederations and petty khanates. Then, starting in the 16th century, the Mongols began converting en masse to Tibetan Buddhism, and the monasteries and lamas slowly gained power over the khans. Finally, in the 17th century, Mongolia was conquered by the Qing dynasty and became a province of China once again, this time very much in a subordinate position.

From the 17th century onwards the borders of Mongolia were eroded as Han Chinese in the south, and Russians in the north, began settling the steppe. Squeezed between them, the modern history of the country has been a balancing act between the two superpowers. With the outbreak of the Chinese Civil War in 1911, Mongolians saw the opportunity to gain independence. The Bogd Khan, a Tibetan lama and the spiritual head of the country, was able to seize control of the northern part of Mongolia (Outer Mongolia), but Inner Mongolia, which had been heavily settled by Han Chinese in the preceding centuries, remained under Chinese control. Independent Mongolia was short-lived, however. In 1919 it was occupied the Chinese. In response the Bogd Khan turned to the colourful Baron Roman von Ungern-Sternberg, a Russian aristocrat with a passion for Tibetan Buddhism and anti-semitism who at the time was running around Siberia with some of the last remnants of the counterrevolutionary White Army. The Mad Baron invaded, and naturally this did not make the Bolsheviks very happy, so they backed a communist uprising. The communists won the three-way war, turning Mongolia into a Soviet satellite state until 1989.

One thing needs to be clear: "Mongolia" as a nation-state is a fairly new entity. Since that region has been primarily inhabited by nomadic tribes until extremely recently and still has a considerable nomadic population, the concept of holding territory is meaningless.

Over the next few generations after Genghis Khans, the Mongol Empire broke apart. A couple of sons and grandsons ruled over the whole thing, but after that, it fell apart. The problem seems to be that the Mongols were a lot better at conquering than they were at holding onto territory. They assimilated into the local (primarily Turkic) cultures. Their biggest problem, though, was that they didn't have a clear inheritance system, so the various khanates succeeding the Mongol Empire often fell apart due to infighting.

One successor fragment, the Golden Horde, took over mostly eastern Europe. It shattered over the next few generations and its successor states got taken over starting in the 1500's by what would become Russia.

One, led by Kublai Khan, became the Chinese Yuan Dynasty. It only lasted a few generations before the Ming Dynasty took over.

I find the Ilkhanate to be the most interesting successor fragment. They went to the Middle East. They killed the Caliph (wrapping him in a carpet and trampling him with horses to avoid shedding royal blood upon the ground) and toppled the Seljuk dynasty. The reason why I find the Ilkhanate most interesting is that, by overthrowing the Seljuk dynasty, it allowed another Turkic dynasty to rise to prominence during its decline. You may have heard of the Ottoman dynasty.

The fourth was the Chagatai Khanate. Its territory was central Asia and northern India. They assimilated with the local Turkic culture. They spent a few generations warring with another Turko-Mongolic empire, the Timurid dynasty (Timur, the last major nomadic conqueror, and his descendants) and hybridized a bit with Persian culture too. The longest-lasting Turko-Mongol-Persian dynasty that would come out of this would be the Mughals, who controlled a large portion of India.