During the time of Genghis Khan, the horde or army wasn't merely a teeming mass of people rushing across the face of Asia destroying all in their path. If they merely teemed across the continent, they wouldn't have had anywhere near the level of effectiveness that they did have. In fact, the army was extremely well-organized and well-articulated, two strengths which, when coupled with the Mongols' speed and skill on horseback, resulted in the formation of the largest land empire that the world has ever seen.
Theoretically, the horde was organized into multiple subunits. It was based on multiples of ten according to the preferred mode of organization for the peoples of the steppes. A ten-man group was called the arvan, a hundred-man group the zuun, and the ten thousand-man group the tumen. The thousand-man group is supposed to have been the level at which the average soldier usually identified himself; the evidence for this comes from the fact that a people of central Afghanistan, the Hazara, look quite a bit like Mongols. The word for one thousand in Persian, which was the dominant language in the area during the Mongol invasions, is Hazara.
While on the move, the tumen had not only the ten thousand men assigned to it, but also their wives, children, posessions, around 40,000 remounts for the soldiers, and a huge amount of small livestock like pack animals, sheep, and goats. It wasn't inconcievable for a tumen to span fifty miles or more when on the march. The livestock would be milked four times a day, which sort of slowed down the progress of the tumen itself. On particularly slow days, the tumen would move no more than five miles or so. Although their livestock slowed the army down, it also provided a portable source of food for the horde, thus greatly reducing the logistical difficulties usually inherent in the military mobilization of an entire people.
The horde was commanded by the Khan, who held court in a portable golden building called a ger. Interactions within this building were strictly ritualized. Visitors always had to stand against the west wall, and the ger itself always faced south. Everyone who came in had to drink fermented mare's milk, called koumiss, and pass through a gauntlet of frowning and potentially deadly guards.
During the time of Genghis Khan, the horde was composed of roughly 60% light cavalry and 40% heavy cavalry, which would be armored with leather so as not to slow the cavalrymen down. The light cavalry usually used an especially strong bow, called the composite bow, to pierce armor and otherwise hurt people. The Mongols rarely used infantry; the closest they came was when they forcibly marched the population of some city's countryside in front of their army as a sort of human shield while they attacked that city. The lucky prisoners got to man siege engines, while the unlucky ones were just used as a screen in front of the main body of the horde for enemy projectiles. Usually, their tactics centered on their ability to fight individually and on their great mobility, which would be expected based on their choice of weapon system. Hit and run was the typical Mongol modus operandi, although the heavy cavalry didn't run as much as the light cavalry since it clearly reduced their effectiveness by making shock combat impossible. The light cavalry of the Mongols was excellent at tricking its opponents into harassing a feigned Mongol retreat before the light cavalry attacked while retreating, picking off the necessarily dispersed pursuers one by one. However, the combined might of thousands of light cavalry missiles concentrated on your army was certainly nothing to sneeze at, either.
The general strategy of the horde was essentially one of terror, especially under the conquests of Chengiz Khan. Aside from their tendency to decimate populations in battle in the manner mentioned above, the Mongols were also very big on mass murder. When the conquered Muslim Khwarizmian empire revolted against their Mongol overlords, murdering garrisons of Mongol troops and reasserting self-rule, the Mongols went on a campaign of extermination. When they captured towns, they had a strong tendency to kill everyone in town and build a huge minaret out of the resulting skulls. The lucky ones were just forced out of the captured city, although most people were killed outright. Any prisoners that were taken were used in siege situations as stated above. One city by the name of Herat was spared, but it unwittingly revolted again after the Mongol army had left, killing the governor they had left behind. When the Mongols had gotten done with Herat after that, only forty people were left alive in the city that had once boasted several hundred thousand.
Very few people were able to withstand Mongol domination. The city of Novgorod in Russia was left alone because of its non-cavalry-friendly terrain, and the Mongols were also defeated in the mountainous Sinai peninsula by the Mamluks who ruled Egypt. The initial Mongol conquest lasted four years and covered the whole of Asia, with the exception of south China, India, and Japan (where they were defeated by the original kamikaze). The Mongols would have probably had Europe, too, having conquered the Polish lancers and the Hungarian horse archers on the Baltic and Adriatic, respectively, but Chengiz Khan died, precipitating an end to conquest for the great horde.
Sources: http://home-4.worldonline.nl/~t543201/web-mongol/mongol-enter-index.htm, as well as Archer Jones' The Art of War in the Western World and Marshall Hodgson's Venture of Islam, Book 2.