I'd probably need a whole post - nay, a couple of posts to explain it. But here's a quick summary:
Both Taiwan, and mainland China call themselves "China." They don't agree on much, but they agree on one thing, that there is but one China. This is called The One China Policy. They just leave it at that.
Taiwan is the Republic of China (ROC). It is a democratic country.
Mainland China is the People's Republic of China (PRC). It is ruled by the Communist party.
In 1911, there was a revolution in China. Dr. Sun Yat Sen overthrew the emperor system, and established the Republic of China (ROC) on the mainland. But the country was unstable. It was carved up into little pieces by European colonialists. Everyone had a piece of China. Then came the Japanese with their colonial ambitions in the 30's. It was a mess.
My Chinese history isn't that great, but my interpretation was that since the country was a mess - all these foreign powers, and local warloards, and the inability of the new government to unite and control the county, there was a communist counter revolution led by Mao Tze Tung.
In the end, it was Mao who ultimately prevailed, and in 1949, established the Peoples Republic of China (PRC) on the mainland. That put the nationalists (ROC), headed by Chang Kai Shek on the run.
They bolted to Taiwan to regroup, with the intention of one day coming back and defeating the communists. Well, that never happened, so the Republic of China settled here on Taiwan.
Taiwan was recognized internationally as the One True China, and communist China were the phonies. In around 1971 or 73 (I forget) the Nixon Administration realized the communists weren't going away, and were getting bigger, so they switched sides. The US used to have military bases on Taiwan, and Taiwan was recognized by the UN. The US withdrew, and the UN kicked Taiwan out in favor of the PRC.
So Taiwan is a sovereign democracy, but has no diplomatic ties with anyone. China makes sure of that. China says Taiwan is a "province" of China, which is complete BS.
So the China in "The China Post" refers to The Republic of China (aka Taiwan).
Reunification with China is a big issue. Most people are fine with the ambiguous current status quo. Others see that Taiwan will eventually and slowly be absorbed by the borg that is China, in the same way it is happening with Hong Kong.
There is also an active separatist movement / party, which is represented by the Taipei Times. They are the "Green" party, but it doesn't have anything to do with the environment. That is just the team's color.
So in the US it is red vs. blue. Here is is blue vs. green. In that way, the structure of politics is the same everywhere.
All of us were ordinary compared to Cynthia Rose. If you set your mind free baby maybe you'll understand.