There were at least two main weaknesses that led to the collapse of the Qing (or Ch'ing) Dynasty (1644-1912). The government was incompetent and corrupt and its military was too weak. These two interrelated problems led to its fall.
Over the centuries, the population of China grew rapidly, putting more pressure on the government to police and care for the population. However, the government did not expand or modernize. Instead, it kept to its old ways which had been more adapted to a smaller population. Thus, as the population grew and social and economic problems resulted, the government was unable to deal with those problems.
At the same time, the weak military allowed foreign countries to dominate China wherever they met. The Opium Wars were the clearest sign of this. This domination by the West disenchanted many Chinese.
Overall, then, the government was too corrupt, ineffective, and weak to police, protect, and care for its growing population. This led to widespread dissatisfaction and the eventual collapse of the dynasty.
The Qing Dynasty of China had a long run, but that run was destined to come to an end during a period of increasing revolutionary fervor across Europe and Asia. A monarchy, the Qing Dynasty, which ran from 1644 to 1912, was ultimately undone by the internal rot that infested many such governing regimes, particularly the Romanov Dynasty in neighboring Russia. Corruption and incompetence had facilitated the development of conditions highly conducive to revolution. The Chinese economy was poorly managed, and the enormous peasantry that characterized China found its already meager existence badly exacerbated by the out-of-control inflation that was rendering what earnings could be had increasingly worthless.
One of the factors that contributed to the Qing Dynasty's downfall was the fact that it was ruled by the Manchu, and China, like many large countries, had its share of ethnic diversity. Manchuria is a part of the nation of China, but its native people are, or were, ethnically distinct from each other, such as the Han, who comprise the overwhelming majority of Chinese. As a minority ethnicity, some mistrust and animosity from the larger ethnicities was almost inevitable, and, as conditions worsened and popular anger at endemic corruption grew, the ruling emperor and his court were increasingly isolated from these other groups.
An important additional factor in the dynasty's fall was its perceived failure to protect China from foreign invaders, including the Portuguese, English, and Spanish, who had colonized much of China and exploited its resources while forcing foreign belief systems, mainly Christianity, upon this ancient Confucian society. The Opium Wars presaged growing and militant hostility towards foreign invaders, and the revolt against the Qing Dynasty during the Boxer Rebellion (1899-1901) represented the opening salvo in the nationalist and anti-monarchical movement that would result in both the dynasty's end and the emergence of a powerful nationalist movement under the leadership of Sun Yat-Sen, revered today in both mainland China and among the ethnic Chinese nationalists who ruled over Taiwan for many years following the Chinese Revolution.
To summarize, the Qing Dynasty's downfall was a result of corruption, growing animosity towards a minority ethnicity in power for too long, growing militancy directed against foreign imperialists, and the fact that was all coalescing during a period of revolutionary fervor across the continent.