In 1236, after years of careful planning, the Mongols attacked Russia. Russia was not a united kingdom - instead, each major city was ruled by a prince, and the princes often fought one another. When the Mongols arrived, the Russian principalities were divided and refused to fight together, making them easy prey for the unified and highly-organized Mongol army. By 1240, all of the major Russian cities had been either destroyed or captured – the city of Vladimir was burned with its population still inside.
In 1241, the Mongols invaded Poland and Hungary simultaneously. Here, too, they triumphed over tens of thousands of European knights and peasant foot-soldiers. Both kingdoms would have been incorporated into the Mongol empire if not for the simple fact that the Great Khan Ogodei (Temujin's third son, who had become Great Khan following Temujin) died, and the European Tumen were recalled to the Mongol capital of Karakorum. This event spared what very well could have been a Mongol push into Central Europe itself; the pope at the time called an anti-Mongol crusade and those Europeans who understood the scope of the threat were terrified of the prospect of the Mongols marching further west. As it happens, the Mongols never came back.
The Mongols were finally stopped militarily by the Mamluk Turks, the rulers of Egypt as of the thirteenth century, who held back a Mongol invasion in 1260. By then, the inertia of the Mongol conquests was already slowing down as the great empire was divided between different grandsons of Temujin; the Mamluk victory did not represent the definitive defeat of the Mongol horde as a whole, just a check on Mongol expansion in one corner of the vast Mongol empire. By then, the Mongol khanates had become truly independent from one another, with Mongol rule eventually collapsing over time (a process that happened in just a few decades in some places, but took centuries in others - Russia was not free of Mongol rule until the second half of the fifteenth century).
Mongol rule had mixed consequences for both Asian and European history. There was a beneficial stabilization in the trade that crossed the west – east axis in Eurasia as a continent, as Silk Road traders enjoyed a relatively peaceful and stable route. It also terrified Europeans, who heard travelers’ tales of the non-Christian “Tatars” in the east who had crushed all opposition, and in Russia it created a complex political situation in which the native Slavic peoples were forced to pay tribute to Mongol lords. To this day, the period of Mongol rule is often taught in Russia as the period of the "Tatar Yoke," when any hope of progress for Russia was suspended for centuries while the rest of Europe advanced; while that may be a bit of an exaggeration, it still has a kernel of truth.