Two Chinas: ROC and PRC | AskHistorians Archive Observer

Taiwan was a relatively recent Chinese territory, but still had a substantial history of Chinese control. China was not the first colonial power in Taiwan - Spain and the Netherlands both built settlements in Taiwan in the early 17th century, with the Dutch defeating Spain in the mid-17th century and becoming the sole colonial power. Chinese settlers had arrived before them, and continued to arrive and settle during European colonial rule. Chinese rule first came to Taiwan in 1662 when Zheng Chenggong (AKA Koxinga) defeated the Dutch and conquered Taiwan, in principle for use as a base for a Ming reconquest of the Mainland from the Qing. In 1683, the Qing took advantage of a violent dispute over the succession for rule in Taiwan and conquered Taiwan, making it part of Fujian province. In the late 19th century, Taiwan became a province in its own right. In 1895, control was ceded to Japan ("in perpetuity") due to China's defeat in the First Sino-Japanese War (the central aim of which was influence over Korea).

Obviously, the Chinese Civil War ended in victory for the PRC and a loss for the ROC, but that is all.

Until the 1990s, both the PRC and the ROC considered the Civil War still ongoing. From the ROC perspective, the time from their "loss" in 1948 to their official recognition in 1991 of the reality that the reconquest of the Mainland from the "communist rebels" wasn't going to happen was the "Period of National Mobilization for Suppression of the Communist Rebellion".

Both the PRC and ROC considered themselves to be the legitimate governments of all of China; both considered Taiwan part of China. There was wide international recognition of the ROC as the government of China, with the ROC hold the UN seat for all of China until 1971.

Both the ROC and PRC planned to bring the territory which had been ruled by the Qing under their governments. This territory included Japanese-occupied territory (Taiwan and the Pescadores, ceded by China to Japan in 1895 at the end of the First Sino-Japanese War, and Manchuria (Japanese-controlled Manchukuo was never recognised by either the PRC or ROC), Japanese-occupied areas of China during the Second Sino-Japanese War, regions controlled by warlords, and regions that had declared themselves independent of China (notably, Tibet and Mongolia (i.e., Outer Mongolia), which had declared independence from China with the fall of the Qing, on the basis that they had been part of the Qing Empire, but not part of China, and shorter-lived independent republics in Xinjiang.

During WWII, the USA and UK agreed to allow the ROC to make Tibet part of China, and in 1943 at the Cairo Conference, that "all the territories Japan has stolen from the Chinese, including Manchuria, Formosa [Taiwan], and the Pescadores, shall be restored to the Republic of China". At end of WWII, the Soviet Union was occupying Manchuria, and the ROC took control of Taiwan and the Pescadores. In Manchuria, the Soviets armed communist forces before and during the withdrawal in March 1946, setting the stage for two and half years of struggle for the control of Manchuria between the PRC and ROC. Taiwan, on the other hand, was securely held by the ROC, and the ROC withdrew there, in principle to use it as a base for the reconquest of the Mainland (very Koxinga-like!). The PRC proceeded to gain control of most of the rest of what it considered "China", taking over Xinjiang (from both the ROC and the Second East Turkestan Republic) in 1949, and then conquering Tibet. The PRC recognised the independence of Soviet-supported Mongolia, and the Soviet Union recognised Chinese (PRC) sovereignty over Manchuria (where previously Russia had many rights by treaty).

At this point, the ROC had greater territorial ambitions than the PRC:

https://en.wikipedia.org/wiki/File:ROC_Administrative_and_Claims.svg

Both the PRC and ROC claimed the territory that they controlled together. In addition, the ROC did not recognise the independence of Mongolia. The ROC finally recognised Mongolian independence in 2002, establishing de facto embassies in the absence of formal diplomatic relations. The ROC constitution continues to describe the Mainland, including Tibet (which the ROC never controlled) and Mongolia (also never controlled by the ROC, nor by the PRC). In practice, these ambitions have been abandoned, but the legal position has not caught up with practicality yet.

Taiwan is divided on the issue of independence vs eventual union with the Mainland. Taiwan has had de facto independence since 1948, and has essentially abandoned its previous claims to rule the Mainland (even if this is still present in the constitution). PRC insistence that Taiwan is part of China complicates any attempt by Taiwan to achieve de jure independence; the PRC has threatened that they "have sufficient abilities to thwart any form of Taiwan independence attempts." PRC official policy remains an offer for Taiwan to join under their "one country, two systems" policy.