/u/MartinGreywolf isn't far off the mark, in my opinion, but misses out on a few factors.
Numbers
I can fully agree on this. The Mongols managed to get together a much larger army than was usual for the period, especially compared to sedentary armies. Important to note is that the Mongols often employed Turkic peoples in their armies as well as their own people. Another thing to keep in mind is that the greatest strength of a nomadic army (Mongol or otherwise) is its mobility, which is gained through the use of multiple horses per warrior. This means the animals can have rest and not die en masse. This also means the nomadic army is limited in its operational range by the availability of fodder, i.e. they are bound to the steppes and the edges of the steppes.
Integration of knowledge
I mentioned above that the Mongols incorporated Turkic peoples into their armies, but these were not the only ones. They also, for example, took in Chinese siege experts, which allowed them to siege fortresses. Another interesting example is the employment of Persian academics (not quite the right word, but the appropriate term escapes me) to create a written form of the Mongol language.
Military organisation and discipline
Nomadic armies were usually well disciplined and coherent, the Mongols are not necessarily special in this. What was special about Chinggis' Mongol army was that he basically created new tribes. Unlike the usual nomadic army, where discipline was high and loyalty to the tribe very important, Chinggis' army used the Tumen system. The army as a whole was divided into Tumen (10,000 soldiers), which were divided into Mingghan (1,000 soldiers), into Zuun (100) and finally divided into the ten soldier units called Arban. If one of the soldiers in an Arban misbehaved, all ten were punished. If an Arban misbehaved the Zuun as a whole would get punished, and so on. Because of this, loyalty and discipline was very high.
The way this helped with the problem of tribal feuds and such was quite simple: the military units became new artificial tribes and the traditional tribes were divided between them.
This organisational model also helped make sure the nomadic armies could take initiative and carry out complex manoeuvres with relative skill and ease.
Sources
Morris Rossabi, Mongols: A Very Short Introduction.
Dennis Singor, 'The Inner Asian Warriors', Journal of the American Oriental Society vol. 101 no. 2 (1981) 133-144.
John Masson Smith, Jr., 'Review: Nomads on Ponies vs. Slaves on Horses', Journal of the American Oriental Society vol. 118, no. 1 (1998) 54-62.
Rocket launchers. No, really.
That aside, this sort of depends on where and when we're talking about. I don't know enough about Mongol rise and wars against Khwarezmian empire, but for invasions of Europe, it was three things.
Numbers. Nomads did sometimes launch sizable (a few thousand perhaps, numbers are either not mentioned or notoriously unreliable) incursions into Eastern Europe (Hungary, Byzantium, Poland), but Mongols managed to gather the biggest army the Europa saw in a while - estimates differ, but Legnica/Sajo campaign had around 50 000 soldiers in two armies.
Discipline. Nomads were usually based around tribal politics, and that wreaks merry havoc with your army organization. Using mercenaries of the same culture against each other was business as usual (e.g. Cuimans in Hungarian service against raiding Cuimans). Mongols, not so much - sure, there were mongols mercenaries, but there was few of them when compared to the size of Mongol armies.
Tactics. Mongols knew how to besiege a city, if they wanted to. This is one reason why Hungarian nobility wasn't all that keen on helping an unpopular monarch with defending the country, they believed that, at worst, Mongols will pillage for a bit and leave without causing any damage to fortified cities. What ended up happening was that only the best fortresses/fortified cities (good position, stone instead of wooden fortifications) managed to hold them off.
To compare them more precisely, we can take Sajo invasion and Cuiman invasion during Ladislaus IV rule (some sources say they were Mongols, they weren't, but it was a popular rumor at the time):
Cuiman invasion was easily fought off by royal army, without the help of most of the nobles, and a great part of royal army were Cuimans settled in Hungary.
Mongol invasion, after killing most of the Hungarian high nobility present at Sajo (palatine, archbishop, several bishops, etc etc), proceeded to pillage the country, burning cities to the ground and taking a lot of slaves. Carmen Miserabile by Rogerius is a first-hand account of escape from Mongol captivity and travel through the lands they pillaged, and the picture painted isn't pretty - people hiding in deep forests or mountains, famine, entire cities empty with streets painted red with blood and so on. Some of it is hyperbole, sure, but the end result was the population of Hungary going from 3 million to 2.