Actually, for the majority of Chinese history, the Great Wall of China was not actually even used. During the Qin Dynasty, the nomads were not a significant threat, and the general Meng Tian led a preemptive strike against the Xiongnu kingdom to the north. It wasn't until the Qin began to crumble that the Xiongnu commenced their raids and harrassment of the Chinese border.

The Han Dynasty started out having to constantly pay tribute to the Xiongnu Kingdon, until Emperor Wudi launched a massive campaign, shattering the Xiongnu and securing China's northern border for the next few hundred years. By the time the nomads became a threat again, the Han did not have the resources to adequately repair and re-garrison the Wall, and attempts were half-hearted. After the fall of the Han, the short lived Jin also made limited efforts to reinforce portions the Wall, but these ended in failure.

During the Three Kingdoms Period, there were again attempts to refortify the northern border with static defense, this time by the Northern Wei against the Rouran Khanganate. This was relatively successful, and the Northern Wei fell to internal factors. The Sui succeeded in unifying all of China, and set to refortifying select sections of the Wall. The Sui also fell shortly to internal factors.

The Tang was the next long-lived dynasty that had unified China. Because the Tang had come from Western China, they had a tradition of horsemanship, and so favored a more mobile defense of the northern border, with armies operating out of fortified bases. Some Tang poetry makes reference to the ruins of the Great Wall, indicating no attempts to use it as a defensive position.

The next major dynasty, the Song, were engaged in intermitant warfare with the Khitai Liao and various Turkic nations. They thus did not have the resources to repair the Wall, and actually used portions of the Wall which had been established during the Northern Wei as the northern border. It is interesting that the Khitai Liao, and later Jurchen Jin (both nomadic peoples) took to trying to refortify the Wall after taking northern China from the Song, which did not succeed to holding back the Mongolians.

Once the Mongols were overthrown, the Ming came into power. As we all know, they are the ones who built the Great Wall as we know it today. This Wall was actually very successful at holding back the nomadic tribes, and only fell once the general Wu Sangui, who commanded the easternmost fortress of the Wall, defected to the Qing and allowed the Manchurians to enter and conquer China. By the time of the Manchurians, gunpowder weapons and the expansion of the Russian Empire made static defense against the nomads irrelvant.

What we see here is that for most of its history, the Wall was only considered a viable defense once the country began to decline. It is important to remember that the relationship between China and the nomads was not actually primarily one of war, but actually of trade, and China was a major provider of food and goods. When the Chinese kingdoms were doing well, the nomads had no need to attack China; likewise, when China was strong, the nomads dared not attack. Once resources became scarce for both sides, the Chinese would turn to the Wall, but would lack the resources to rebuild and refortify, or could only do so in specific regions.

The fact that the Wall was mostly ineffective is actually partially why Xinjiang is part of China today. Xinjiang produced some of the finest horses in the region, so Han Emperor Wudi made certain to control the region; subsequent emperors would also try, notably Taizong of the Tang. With these horses, the Chinese were capable of mounting campaigns against the nomads, and was regarded as a preferable means of defense than maintaining the Wall.

Also interesting to note is that the Mongols were only a major threat during a few centuries of history. For the most part, it was various Manchurian tribes and Turkic kingdoms that were attempting to break through the Wall.

Tl;dr The Mongols were not really the main group trying to pass the Wall, and when they did, the Wall was not in any shape to pose any obstacle. Mobile defenses were preferable to maintaining the Wall, and even under the Ming, the shorter parts of the Wall were not meant to stop the invaders so much as to serve as watch-posts so that the army, based in heavily defended fortresses, could respond to the attack.

The Mongols did not fight Song China along the Great Wall, which China had already lost during the civil wars and nomad invasions at the end of Tang dynasty in 907 AD, Song China had never had control of the Great Wall.

The Mongols simply circumvented the segmented walls during their invasion of the Jurchen Jin dynasty who left the great wall largely un-maintained. It wasn't until Ming dynasty did the Chinese finally linked all segments of the Great Wall as it is seen today.

Follow-up question - if they didn't, was it because they couldn't bring their horses with them?