It depends on how much you believe in the butterfly effect.
From a technological standpoint, the Mongols were the first empire and people to use gunpowder outside of China. They brought bombs with them to Japan and might have used them in Russia and Hungary at some point although they have been described as naphtha weapons as well. This is unclear because early Arabic accounts of gunpowder weapons construed them with an early incendiary - naphtha. The Mongols definitely used gunpowder weapons in China, although this is not exactly new since gunpowder had already been used in the previous wars between Jin and Song.
From a knowledge and travel standpoint, the Mongols were crucial to the spread of knowledge, although again this can be quite unclear because the Mongols themselves left few primary source documents. This is quite understandable if you realize that up until the point where Genghis Khan adopted the Uyghur script for the Mongols, the Mongols essentially had no written language, so I would presume they were all illiterate. Even the Secret History of the Mongols was written in a Chinese transcription of Mongolian, and it probably wasn't an original. Scholars think the first ones were written in Uyghur. So imagine if English was written in Russian, but then rewritten in Arabic, phonetically. Almost everything we know about the Mongols comes from second source accounts such as their subjects, vassals, and travelers like Marco Polo. Under the Mongols travel between the different continents became safer than I probably any other time before it because up until that point, there had never been an empire that big under one single power. This was actually a really brief period lasting until Genghis' death, so like - maybe a decade or two at most. The Mongol Empire under its separate appanages were still considerably safe because they all paid respect to the "Supreme Khan" who was ostensibly situated in the east, holding the empire of Yuan dynasty. The Silk Road(s) reached their greatest extent during the age of Mongol Empire.
Marco Polo was the most well known of the traveler traders who took advantage of the Pax Mongolica, or more simply known as Mongol Peace, but he wasn't the only one. Less known are Giovanni da Pian del Carpine, Giovanni de' Marignolli, Odoric of Pordenone, John of Montecorvino, William of Rubruck, André de Longjumeau, and not to mention Marco Polo's father and uncle Niccolò and Maffeo Polo. Without guarantee of safe passage, reaching such far away destinations such as Beijing would certainly have been impossible if nigh impossible for Europeans at the time. From east to west there were Isa Kelemechi, Rabban Bar Sauma, and Yahballaha III. To get an idea of how connected the world was at this point, Andronikos the second of Byzantium was technically one of Kublai Khan's in laws because he had some half sisters who married Genghis Khan's great grandsons. I'm not sure there was any other point in time when you could claim that someone in China was somehow connected by political marriage to a major Christian medieval ruler in Europe.
From a political standpoint, many of the polities we know of like China, Russia, and India might simply not exist. Without the Mongols Russia as we know it would probably unrecognizable. Certainly the Mongol influence was a major factor in the rise of Moscow. China's capital may not be Beijing today and the city might not even have gained the significance it did without Mongol patronage. Tibet also came under China's suzerainty for the first time during the Yuan dynasty. The Ming dynasty was a result of the overthrow of the Mongols. So was Korea's last dynasty the Joseon. The Timurids would not exist, nor would most of the Mongol successor states in eastern Europe such as the Crimean Khanate. Without Timurids, no Mughals which were instrumental in Indian history. I've heard it said that Columbus' voyages at least in part inspired by Marco Polo's tales, so those might not have happened. Persia as we know it would be pretty different. No Ilkhanate to speak of.
There would be way less people with the last name Khan.
To sum up:
It was the largest human polity up up. They were the first non-Chinese to conquer China. The first and only people to conquer both China and Persia. The first people to simultaneously set up empires in China, the middle east, and Europe. Militarily the Mongol armies conducted warfare successfully on several fronts across all Eurasia and managed to achieve victory on a level never before seen. They were the first people to use gunpowder weapons outside of China and arguably the first people to use guns as well. Nearly all the earliest specimens of cannon are dated to the Yuan dynasty and the Heilongjiang hand cannon is the oldest extant dated firearm in the world. It was used by the Yuan dynasty to put down an insurrection by a Mongol prince in 1288. There are proponents of the Mongol spread of gunpowder throughout Eurasia, but this is an unproven theory and there is no hard evidence to back this up. The Mongol Empire would not be surpassed in size until several centuries later during the height of the British Empire.
Probably at no other time was Eurasia so safely connected. Within the span of a century at least several Europeans traveled as far as China whereas zero Europeans had ever been to China before. Many pilgrims from China were also able to visit places in the middle east and Europe, going to Paris and Rome. Raban bar Salma is the most notable of them. Countless unnamed people were no doubt displaced and forced to move with the Mongol armies, such as the Chinese and Muslim engineers. Syriac and Afghani siege engineers visited places as far to the east as Xiangyang while Chinese engineers might have gone as far west as Russia and Hungary. Muslim communities were set up in China and were possibly instrumental in the construction of the Chinese Hui community who trace their lineage to a Muslim administrator the Mongols set up. I like to think of this period as a sort of preview of the globalization movement.
Politically the Mongols were intrinsic to the national formation of several modern countries. Mongolia is kind of hard to imagine without Genghis Khan. Russia and China would arguably be significantly different from a geographical standpoint since Moscow and Beijing might not have gained pre-eminence over their respective regions. Tibet might never have been considered a part of the Chinese empire. India too was significantly affected by the Mongol successor state of the Timurids and Mughals. They were a fundamental force of the central asian influence on India and also played a role in the rise of the British Empire.
The Mongol and Turkish word Ordu denoting some form of military unit or establishment related to the khan is the etymological ancestor of Horde. Without the Mongols and the Turks the Horde in World of Warcraft would probably be called something else.