Taiwan is a province of China as the Japanese, who occupied Taiwan until 1945, had to give it back to China. China used to be an empire until 1911 and after was ruled by KMT until 1949, when this party has been defeated by the gong chang dang (共产党 or Communists). It was Chiang Kai-shek who considered himself still as the legitimate leader of all Chinese, but in exile on a Chinese province, called Taiwan. His "Republic of China" did not mean Taiwan as a province, but basically the mainland China, that he wanted to win back either militarilly or peacefully.
Just keep in mind the two German states that remerged together into one single state after the wall of Berlin was destroyed in 1989.
The problem between China and Taiwan is mainly the problem between two dictators, Chiang Kai-shek with his KMT on one side and Mao Zedong with the Communists on the other. Both always considered there were only one China and Taiwan was considered by both as a Chinese province.
Now there is within Taiwan a new force of mainly Taiwanese born who had no real ties with mainland China, but they are opposed to the KMT refugees and still are in minority.
Officially, Taiwan still is a province of China and the "Republic of China", that was mainland China's official name between 1912 and 1949 continued to exist on exile, while the mainland changed its name in "Peoples Republic of China". Google Map described Taiwan properly by international legal standards, as Taiwan officially still is a province!

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