A bunch of Han Chinese moved to the island. The Dutch established a small colony there. Japan took over. Post-WW2 they were given independence from Japan. The pre-communist leader of China moved to Taiwan and took over politically. The KMT party ruled until recently, when Taiwan peacefully transitioned to a democracy and now has two main parties.
There is nothing in Taiwan under Chinese control. They have their own government, constitution, military, their own passports that are recognized worldwide, etc. However, there is a looming threat that China will attempt to take over which drives mandatory military service, and tight relations with the US.
Both PRC and ROC say that Taiwan is entirely under Chinese control, that there's only one China, and that they govern all of its territory.
Obviously, Taiwan has no plans to reconquer the mainland any time soon. However, PRC China threatens that if Taiwan formally declares independence it's back to a shooting war.
So yes, Taiwan still maintains governors for the areas of the mainland (last I heard they're all very old as they obviously are not standing for reelection back in their home district). This surprises people when they first learn of it.
It seems to a lot of outsiders like it is more about preserving the very carefully kept status quo than it is about a serious territorial claim.
But I don't really know that much about it. This topic tends to bring out strong opinions in people on one side or the other and I haven't come across many people who do know both sides who will chat freely.
Taiwan is a democracy, and one of the major parties firmly opposes any concept of Chinese rule and seeks recognition of Taiwan. The other party leans towards being the sole leader of all of China. However in practice (being currently elected) they just improve economic ties with China while opposing unification or formal recognition of Taiwan.
The PRC sees Taiwan as a province of China that they should rule, as you mentioned.
FYI the ROC also sees Taiwan as a province of China. There is another province called FuJian province, ROC.
http://en.wikipedia.org/wiki/Fujian_Province,_Republic_of_Ch...
Already been one war fought along that border.
It's also worth noticing that India was never an integral country in the past.
The case of Taiwan is even more confusing. A world globe I have from a decade ago, commercially published by a private business corporation rather than officially published, shows Taiwan and the mainland territories of China in the same color. (Usually this globe distinguishes different countries by different colors.) However, the globe also marks the location of both Beijing and Taipei (so spelled) with a star symbol indicating a capital city, suggesting that Taipei is the seat of a national government. United States law under the Taiwan Relations Act strikes a delicate balance between agreeing with the original assertion of both the P.R.C. and R.O.C. regimes that there is one China and Taiwan is a part of China and the current facts on the ground that China (the P.R.C.) and Taiwan (the island of Taiwan and various outlying islands, including some that historically were part of Fujian Province) are under distinct national administrations.
The Wikipedia article on Taiwan independence
http://en.wikipedia.org/wiki/Taiwan_independence
seems to have enjoyed some good editing back-and-forth so that it is not entirely slanted to one point of view or another.