In advance - sorry for possible typos and sloppy English

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First thing that will help a reader to understand nuances of Mongol military machine is understanding its differences from other nomadic forces, which tactics Europeans (and anybody else, really) utilized against them and why this didn't work against Mongols.

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Nomads were huge existential threat to sedentary civilizations from West Europe to China for a very long time, mainly because of their great skills in horseriding, ruthlessness (sparked from harsh living conditions in the steppes) and skillful utilization of tactics. Nomads of different cultures and origins were uniformally renowned for their agility, mastery of bow and ability to insantly appear out of nowhere, plunder frontier lands of their neighbours and just as quickly disappear into nowhere. Despite their often initial successes and painful defeats inflicted upon their enemies, nomads weren't invincible: they often lacked proper resource base and sometimes technologies to score decisive wins against sedentary empires. In the long run, nomads usually find themselves to be pushed back, subjugated or outright destroyed by their enemies learning new tricks to combat them (with China and argiably Rus Kingdoms having considerable success in raiding into the steppes and eradicating nomadic settlements there).

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Different strategies adopted by sedentary kingdoms to reverse tides of war, often included active use of horse archers themselves (usually mercenaries or vassals from other nomads which in turn would teach archery to their new masters/allies, giving them advantage), building of long defensive lines (great Wall of China, Kievan line of border fortresses) to remove nomads' ability of freely passing into your lands undeterred and combined arms warfare. Supported by strong cavalry detachments, other parts of army, including infantry, were surprisingly effective at establishing footholds in unknown territory, repelling enemy attacks and taking rich lands away from nomads, eventually putting them on the brink of defeat. It also turned out that massed cavalry armies weren't really effective at taking castles and cities: besieging and starving them out - yes, but in case they weren't able to breach defences it was possible for the besieged to hold enough for the reinforcements to arrive.

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Chinese achieved great success in utilizing combined arms warfare to slowly, but steadily take hold in frontier regions. Romans would often bide their time and wait for proper weather conditions in order to strike nomadic incursions at their weakest (they did repel some Sarmatian invasions in 2nd century AD by attacking them during spring mudseason when horses were bogged down in difficult terrain). Other countries such as Rus Principalities primarily switched to cavalry armies themselves, adding more heavily armed shock troops supported by mercenary/newly formed horse archers to crush enemy lines.

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Mongols, however, were different from other nomads. Ruled by very strict and able leaders, they were very quick to adapt to sedentary inventions themselves, learning on the fly and employing new types of weaponry when needed. Mongols utilized very effective strategy of firstly engaging and utterly destroying enemy armies in the field (seeking aforementioned engagement themselves on favourable conditions) to remove any possible deterrance to their campaign in the long run. With mainstay enemy forces gone and nobody to oppose them, Mongols would swiftly move in and cut enemy supply and communication lines, isolating towns and fortresses from other parts of country. Bringing in war machines (which they learned on how to build and operate from the Chinese, Persians and Arabs), they would eventually breach defences after some time and annihilate now outnumbered defenders. Very effective and smart strategy, and one impossible to fullfil without level of discipline achieved by Mongols - a feat arguably unmatched by other nomadic empires. By establishing steady recruitment of fresh warriors in their ranks and constant training sessions (collective hunts on agile steppe prey, which would teach recruits how to properly act in unison with their comrades) Mongol Empire reached level of effectiveness far outmatching other nations of its era.

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Medieval period is known for its constant warfare among petty kingdoms, and it was another reason why Mongols were so effective: their opponents were simply pretty politically unstable and weak from military standpoint. Small kingdoms mean small armies which are very easy to track and destroy. Lack of unity among nobles means it is easy to turn them against one another or at least expect them to coordinate rather poorly. Feudal nations (especially European ones) lacked technological supremacy of Antiquity, and their main effective fighting force (mounted knights supported by their retinue and some amount of levy soldiers) was very vulnerable to numerous and mobile horse riding opponents. Choosing to perfect their individual martial skills, knights weren't keen on actually utuilising complex strategies, often preferring to charge their enemy head on, which was deadly trap against smart opponents. Yes, knights were well armed and unmatched in terms of direct combat, but they were few in numbers and could be easily cut off from the rest of their host (due to their unruly nature and habits of overpursuiting). With their main fighting component gone, medieval European (or many other) armies would be easily routed and put to flight, and superiour Mongol mobility made sure none of them would escape to safe places.

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This situation gave Mongols huge edge over sedentary armies of their era. Their opponents lacked proper and able fighting infantry or field war machines (such as cannons) and their own cavalry was quite small in numbers. Mongols' discipline also ensured they would usually trump any other horse archers. Mongol generals knew of their enemies' weaknesses and exploited them with impunity: such as at the battle of Kalka river in 1223 when Mongol vanguard suddenly attacked and put to flight weaker, nomadic part of Rus princes' army. Fleeing horse archers disorganized other parts of the army, allowing Mongols to quickly encircle them and rain arrows upon them, ensuring victory. At the battle of Sit river Mongols, due to their superiour scouting, knew exact location of the Rus camp, and it helped them in surprise attacking and destroying it.

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Of course, Mongols weren't invincible and had their fare shares of defeats. Big Mongol host was annihilated at Parwan in 1221, and Mongols also lost control over parts of their Near East territories after disaster at Ain-Jalut. Even if they won some engagements, it would sometimes lead to them suffering big casualties. For example, at battle of Indus river in 1221 Mongols nearly snatched victory out of jaws of defeat, and during siege of Kolomna at least some parts of Mongol frontline forces were destroyed, and Rus' forces were able to kill one of their commanders. Despite these failures, Mongols still won multiple victories which negated their rare defeats, and were able to, at the very least, outlast their opponents due to effective supply machine.

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Despite their very high combat efficiency and staggering list of conquered regions, Mongols couldn't adequately maintain their empire in the long run. Unified Mongol state wasn't able to govern all the conquered territories (and possible unruly populace or appointed officials staying there). After their undisputed rulers died, Mongol warchiefs were quick to start the infighting which led to their empire collapsing and fracturing into several, much less potent, dynasties. With their huge resource base and ability to support long supply/recruitment chains gone, Mongol leaders were unable to maintain their territories at peak performance. With pillars which supported colossus of the nomadic army now gone, their military forces started to degrade over time (which wasn't helped by them constantly waging wars against each other). With discipline and military equipment becoming much less advanced in comparison to their neighbours (and their neighbours eventually far surpassing Mongols in these ares as their military science started to advance in huge leaps in later centuries of sedentary-nomadic conflict), Mongol successor states lost grasp over their subjects eventually, were pushed out of formerly conquered territories and faced miltitude of defeats.

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In short, European armies of 13th century weren't on par with Mongols to effectively oppose them in the field. But with passing of time Mongol successor states became weaker and their opponents learned much, which eventually led to them being in turn divided, conquered and incorporated into other empires.

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My primary source of information on the subject is:

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Roman Khrapachevsky - Military State of Kenghis Khan.