Long story, but here goes.
First and foremost, there is no such country as “Taiwan”. There is a country called the “Republic of China” that people call Taiwan and goes by other names like China-Taipei.
Now, for all intents and purposes the Republic of China is a “nation”. It has exclusive control over a territory (the Island of Taiwan, plus some other islands), it has a permanent population, and it has a government.
What it is not is a “sovereign nation”. In other words, only about a dozen other sovereign nations have diplomatic relations with the Republic of China. The rest treat it as a breakaway province of the People’s Republic of China. In other words, if you look at, say, the United States, their official position is that Taiwan is part of the PRC even though the PRC doesn’t control it. This is generally referred to as the “One China Policy”.
But it wasn’t always this way. From 1949 to 1974, the Republic of China was seen as the legitimate government of all of China. The Chinese Mainland was a breakaway province of the RoC. The PRC government in Beijing was not legitimate. Countries like the Soviet Union and other nations in the Eastern Bloc immediately recognized the PRC, but the majority of the world’s governments recognized the RoC as the legitimate government of China.
When people say “Taiwan”, here’s what they think the Republic of China is:
You can see that little island Quemoy is orange too. It’s part of Taiwan and the RoC.
But if you ask the RoC what it is, here’s what they will tell you:
The Republic of China is Taiwan, the PRC, most of Mongolia, plus parts of Burma, Bhutan, India, Japan, Afghanistan, Pakistan, Russia and Tajikstan.
By 1974, the United States and most of the world’s other nations pretty much conceded that the RoC was not the legitimate government of this territory. They recognized that the PRC had the authority to deal with these territorial claims and recognized that it was the government of the “One China”. The PRC got the seat in the United Nations that used to be held by the RoC.
The relationship between the PRC and the RoC is complicated. Officially, they have no relationship and each claims the other is an illegitimate government of the territory they control. Unofficially, the PRC would like to take the RoC back into China even if it means treating it with special status, like Hong Kong. The RoC has no interest in this arrangement.
So why doesn’t the RoC declare independence? Two reasons - they don’t want to and China has enough diplomatic clout to prevent it from being recognized. If Taiwan declared independence, that would be an acknowledgement that the PRC government is entirely legitimate. The PRC would diplomatically punish any nation that recognized Taiwan independence.
So, for now, both are happy with the status quo. The RoC continues to claim all of China, and no other nation officially recognizes it, and China continues to claim Taiwan, even though they have no authority there.