Traditionally, much of the blame for the collapse of the Ming was placed on the Wanli Emperor, who ruled from 1572-1620. He was only 10 years old at the start of his reign, and the government was run by a regent for the first 10 years of his reign. The regent, Grand Secretary Zhang Juzheng, restored the treasury to a healthy state, by methods such as avoiding expensive wars. He also attempted long-term financial reform, less successfully - some of his plans simply failed, while other were reversed following his death in 1582. From 1582 to 1600, the government was faced by expensive wars again, with two major rebellions and the Japanese invasion of Korea (the Ming aided Korea because (a) Korea was a tributary state and ally, and (b) China was Japan's next target, so better to stop the Japanese in Korea before they got into China). These wars largely emptied the treasury.
The last 20 years of Wanli's reign are why he is blamed for the fall of the Ming: he essentially retired from government (without abdicating), leaving much of the administration in the hands of eunuchs. Unfortunately for the Ming, the last 44 of Ming rule (1600-1644) were a time of much drought and flood (leading to widespread famine), plague (e.g., one outbreak killed about a quarter of the population of Beijing) and war. This new and last round of war for the Ming began with Manchu/Qing expansion into the Ming's territories north of the Great Wall (and some Manchu/Qing and Mongol raids across the Great Wall). The poor state of the government's finances meant that little could be done to help those suffering from famine and plague. Because of this, Wanli, who reversed rather than continued Zhang's financial reforms, and emptied the treasury through war, was blamed for the inability of his three successors to help the people, and in true Chinese style, the official history of the Ming (written after the Qing takeover - it was traditionally the new dynasty's duty to write the official history of the previous dynasty) blamed him for the fall of the Ming. Chinese political theory of the time justified the change from one dynasty to another by claiming that Heaven favoured the new dynasty due to bad behaviour by the recent rulers of the old dynasty.
Other modern explanations for the difficulties faced by the late Ming include a collapse of the silver supply, due to a drastic reduction in the flow of American silver, making the economic problems worse (taxpayers, who paid taxes in silver, found their taxes effectively much higher due to the higher price of silver resulting from the shortage of silver), and blame the drought and floods (and generally poor crop yields) on the Little Ice Age.
Whether one blames Wanli, the silver shortage, or the Little Ice Age, there was much suffering in China during the last decades of Ming rule. As suffering peasants seeing no help from their government sometimes do, many of them revolted, and the last Ming emperors faced much rebellion, including two major rebellions. These rebellions further disrupted the economy, and distracted the Ming from stopping the Qing, who made further progress into Ming territories north of the border.
1644 saw the Ming in poor shape. North of the wall, there was one powerful Ming army, led by the general Wu Sangui, defending Ningyuan (today Xingcheng in Liaoning province, administered by the city of Huludao). Sichuan fell to rebels under the leader Zhang Xianzhong, and worse, another rebel leader, Li Zicheng, who held Xi'an, declared himself the ruler of the new Shun dynasty, and marched on Beijing. Wu Sangui's army was the only force that could stop them, and he was ordered to abandon Ningyuan (and therefore China north of the wall) and head for Beijing. Starting 450km away, failed to get there in time, and Beijing fell to Li Zicheng's rebels without a battle, with the "last" Ming emperor hanging himself (from a tree which was afterwards called "the guilty pagoda tree"). Other Ming emperors were crowned and continued the fight against the Qing for many, many years, but they're not considered "real" emperors by history.
Wu Sangui was now at Shanhai Pass, at the eastern end of the Great Wall. The Qing army, led by Prince Dorgon, regent for the 7 year old new emperor (the previous emperor, one of Dorgon's older brothers, had died the previous year, and Dorgon had strongly opposed an adult son of the former emperor being crowned - a child emperor would give him the chance to be regent, and regent he was until his death in 1650), had followed Wu to the Great Wall. He was now between the rebels and the Qing. Li Zicheng sent a messenger asking him to defect, offering him money and a noble title. Wu considered this offer for a few days, and decided to accept it. However, while leaving his army at pass, and heading for Beijing, he learned of the predatory destructiveness of the rebels, and the arrest (and soon, also execution) of his father, he changed his mind and returned to the pass. Wu's forces were sufficient to hold the pass, stopping two rebel attempts to take it, but was too weak to recapture Beijing. His solution: ally with the Qing. Thus, Wu and the Qing headed for Beijing, with Wu spearheading the advance. They took Beijing, and the Qing Dynasty became the official rulers of China.
The next step for the Qing: conquer the rest of China. With the disruptions caused by rebellions, and the collapse of the Ming government's finances, and further defections of Chinese forces adding to their armies, the Qing were able to achieve their goal of conquering China by 1663, a mere 2 decades after the Qing capture of Beijing. In 1662, Wu Sangui, still fighting for the Qing, cornered and killed the last of the still-resisting Ming "emperors". The following year, Qing forces pushed the last major pro-Ming forces off the mainland (these retreating Ming loyalists, led by Zheng Chenggong AKA Koxinga, proceeded to take Taiwan from its Dutch colonial rulers, planning to use it as a base for the reconquest of the mainland).
Qing rule was still not secure. Three major Chinese military commanders (including Wu Sangui) had been given large territories to administer in southern China as a reward for their loyal service. They wanted their sons to be able to succeed them, but the Qing didn't want those territories to be hereditary, and in 1673 they revolted against the Qing. This, the Three Feudatories Rebellion lasted until 1681. With this revolt suppressed, the Qing turned to Taiwan, and took advantage of a succession dispute among Zheng Chenggong's successors to take Taiwan in 1683.
In summary, the Ming collapsed due to famine and economic collapse driving revolts against them, with that economic collapse hampering the Ming response. When Beijing fell to Li Zicheng in 1644, the Qing had the strongest army in northern China, but their takeover of China didn't automatically follow. The critical step was Li Zicheng driving Wu Sangui to ally with the Qing - a serious (and fatal) political error by Li. Economic and environment causes were very important in the fall of the Ming, along with Ming failure to reform government finances (for which the Wanli Emperor was blamed), but in the end, the actions of individuals mattered: Li, Wu, and Dorgon.