Regardless of territorial history, the PRC's claim of Taiwan stems from the civil war that had begun in China before Japan invaded in the 1930's, and resumed when war with Japan ended.
The forces were the Kuomintang, the government formed by Sun-Yat Sen during the Xinhai Revolution of 1911 when the last Qing dynasty was overthrown and the colonizing European powers evicted. During WW2, the Kuomintang government was led by Chiang Kai-Shek.
And the communists under Mao Tse-Tung. Obviously, the communists prevailed. Chiang took his remaining forces and government, and those citizens closely allied with the Kuomintang government (who probably would have been shot by the communists), and moved en mass to the large island of Taiwan, which had been a Japanese territory, taken from the Qing dynasty by Japan during the 1895 Sino-Japanese war.
As the early PRC had no real amphibious war capability, the fleeing ROC (Republic of China) government was able to consolidate its position and military strength, to where an invasion of Taiwan would be very costly.
For a long time after that war, the ROC maintained a seat on the UN's security council, as the representative of the Chinese people. That was lost when the US and the PRC warmed relations in the 1970's, and the PRC was given the security council seat.
So, technically speaking, the Chinese Civil War of 1949-1950 is still going on, as the Kuomintang government was never fully defeated, and still exists (and thrives) today.
Viewed in the context of how the two nations came to be, the PRC's claim isn't so much for the land, but authority over the Chinese people who live on that land, and especially final defeat of the Kuomintang government after almost 70 years.
Interesting perspective on this situation today: Many successful Chinese businesses are actually owned and operated by Taiwanese citizens. One reason: Citizens living in the PRC, which had been closed to the world for decades, had little experience in international business and finance when China began to open up economically to the rest of the world, while their counterparts on Taiwan had been selling goods to the western nations for a very long time.
The PRC maintains a 'one China' policy, that there is only one Chinese government and that's them. They are willing to look the other way for very successful Taiwanese owned businesses operating in China, like Foxconn, who makes a lot of electronic equipment for Apple.
Not upsetting their productive business partners and throwing major employers into disarray is one reason the PRC would be very reluctant to attack the ROC today.