There are a number of tactical and strategic reasons that the Mongols were successful.
- Core of strong leaders:
Not only were the upper levels of military leadership strong, but the mid-level and lower level leadership was also very strong.
- Flexibility of tactics:
They used whatever means necessary to defeat their enemies, including using direct confrontation by large forces, using of smaller forces in guerrilla tactics, deceit and subterfuge, etc. During an attack, local commanders had a broad outline of the campaign strategy, and also had the freedom to achieve their tactical goals as they saw fit.
A common tactic was to intentionally tire their opponents horses. First they charged into battle, then feigned retreat, enticing their opponents to pursue. Once their opponents' horses were tired, the Mongols would switch to fresh horses, and return and massacre.
It is important to note that the Mongols very much preferred their opponents to surrender, and most conquered areas came to be controlled in this way.
Mongolian made crossbows and bows were the best in the world at that time. They were composed of compound materials and took a number of years to manufacture.
They additionally kept Chinese engineering units in the field for events of siege warfare.
- Public Relations:
Mongols treated surrendering populations very well; as they economical favored merchants and poor farmers over elites, the general population was well disposed towards them. Conversely, the Mongols "made examples" of opponents who fought back, and did not hesitant to brutally torture and kill those who took the field against them.
They additionally deceived commoners and merchants about the size of their army; while it was modestly sized, they led their opponents to believe that they were had an enormous fighting force (a 'Horde'). Not only did this intimidate their enemies and the surrounding populace, it also led to tactical errors on the part of their enemies. Specifically, enemy armies expected many unskilled fighters, and instead they encountered smaller, sophisticated fighting forces.
- Military Intelligence Mismatch:
The Mongol data collection methods were very strong. As merchants benefited greatly from Mongol expansion, merchants eagerly provided information about the Mongol's enemies. As merchants knew the ruling classes and geography of to-be-territories, the shared information was of great value.
The Mongol opponents, however, knew primarily what the Mongol PR told them: that in surrender they would be treated well, if they fought they would be not just defeated but annihilated, and that the Mongol army was enormous.
- Strong economic growth in conquered areas:
The Mongols replaced the ruling elites in conquered areas, subsequently lowering tax burdens on farmers, artisans, and merchants. They also encouraged trade by establishing safe, easily travel-able trade routes across Asia, allowing safe trade for the first time in history between China, East Asia, and Persia, the Muslim world, and Europe. As they controlled a huge area, their lower taxes resulted in a huge revenue stream, to allow very strong logistical support.
- Supply and Logistics:
The Mongol forces stayed well supplied by both living off the land, maintaining a relatively small fighting force, encouraging merchant traders in conquered areas, and a large number of horses for direct logistical support during campaigns.
They also had very strong communication between forces, using relays of riders.
- Espirit du Corps:
The fighters in Mongol units were very tough men - they grew on the plains as nomads, and were accustomed to thinking independently and solving problems, developing strong endurance while traveling with excellent horsemanship, while having a cultural tradition of respect for wise authority.
In summary, Mongols were a very well organized, strategically and tactically strong fighting force, with a number of reasons for extensive success.
Sources: While the wikipedia article is ok, I really respect entry in the Encyclopedia of Military History by Dupuy. They provide a very non-biased, factually based account of Mongol success. The anti-mongol propaganda, developed by the conquered elites in China, SW Asia, and Eastern Europe is definitely skewed, and still skews modern Mongol History until even today. In this regard, Dupuy did a fantastic job in conveying why the Mongols were successful conquerors.
Your map shows primarily the Mongol conquered territories during Genghis Khan's time. Tamerlane, who was also a Mongol, conquered India and set up the Mogul empire, which through its effective administration taught the British economics, war, and management of subject populations, allowing them to eventually build their empire. In this regard, the British could be viewed as successors to the Mongol.
The trade that flourish during Mongol rule gave western Europeans a taste for the luxury goods of East Asia; when the empire collapsed in the late 1300's, prices went up due to disruptions in trade, and the desire by European elites for luxury goods led to the beginning of colonialism and European expansion.