There is only one China. China includes mainland China and Taiwan. There are two political entities that claim to be the rightful government of all China (both mainland and Taiwan): they are the People’s Republic of China (PRC), and the Republic of China (ROC).
The People’s Republic of China has effective control on mainland China, but also claims sovereignty over Taiwan. The Republic of China has effective control on Taiwan, but also claims sovereignty over mainland China. Taiwan is considered a rebel province by the People’s Republic of China.
Internationally, most countries and the UN recognize the People’s Republic of China as the only legitimate government over China (both mainland and Taiwan). Many countries, however, recognize the de facto control of the Republic of China over Taiwan, and are willing to subscribe agreements under the understanding that they are not formal diplomatic agreements within nations.
The People’s Republic of China tolerates the existence of the Republic of China because
- It does not threat the integrity of China. ROC does not pretend to declare Taiwan as independent, so they are not officially breaking China.
- The ROC cannot overthrow the PRC. ROC in Taiwan does not pose a threat to the existence of the PRC.
- The ROC has a powerful de facto ally that, although they do not recognize the ROC as the official Chinese government, nor as the de facto government on Taiwan, it will defend the ROC on an eventual takeover by the PRC.
So, while de facto there are two governments on two different territories (which means that de facto Taiwan is an independent country from mainland China), de jure it is only one country with two competing governments, and only one of them internationally legitimate.
And, so far, it suits to all interested parties. (The moment the people on Taiwan vote that they want to make de jure their de facto independence, things will get ugly.)