Taiwan (Republic of China) and China (People's Republic of China) are different governments that both lay claim to the same territory.

The TL;DR is that in 1949 the communists won the Chinese civil war and the remaining nationalist opposition retreated to Taiwan, beginning the state of affairs that we have today.

PRC considers Taiwan part of its core territory and will not renounce its claims. RoC has, since 1991, officially recognised that they can't retake the mainland, but there's ongoing debate about whether or not Taiwanese reunification or an independent Taiwan is the end state.

Not according to Taiwan. Taiwan sees itself as the legitimate China, with territorial claims on the mainland.