Is Taiwan part of China or not?
At present, the nation of "China" is represented by two separate authorities: the Republic of China (commonly referred to as Taiwan) and the People's Republic of China (in mainland China and commonly referred to as just China). During the aftermath of World War II, infighting on the Chinese mainland resumed, with the Republic of China government (established in 1912 as the successor to the Qing Dynasty) relocating to Taiwan, and the group led by Mao Zedong (毛澤東) creating the People's Republic of China. To this day, the PRC has control over mainland China, while the ROC has control over Taiwan island, the Penghu archipelago, and minor islands Kinmen and Matsu, while each government claims sovereignty over the whole of "China".
Simply put, the PRC claims Taiwan to be part of the PRC as it is the successor state to the ROC (which it views as losing the civil war), and the ROC views the PRC as an illegal state occupying China.
As each government claims there is only one China (known as the 1992 Consensus), pressure has been put on the international community to choose whether the PRC or ROC is the sole representative of the Chinese nation. At present, the large majority of the world has official diplomatic relations with Beijing while still retaining economic and cultural ties with Taipei. Please be aware when visiting Taiwan you will not have embassy or consular protection unless you hold the passport of one of the 21 countries that have diplomatic relations with the ROC. That being said, most countries have economic and cultural exchange offices which serve as un-official embassies, providing most consular services. Countries such as the United States provide consular services in Taiwan through a non-profit organisation called the American Institute in Taiwan (美國在台協會). AIT officials are appointed by the US State Department in a manner similar to other embassy staff, according to the Taiwan Relations Act which categorises Taiwan as a country for US government procedures, and provides defence assistance in the event of a military altercation with the PRC. Abroad, the ROC is most commonly represented by the Taipei Economic and Cultural Office (TECO or TECRO) and is commonly referred to within international organisations as "Chinese Taipei" at the behest of the PRC government.
So, back to the question of whether Taiwan is part of China or not.
Taiwan is part of its own definition of China under the ROC with Taipei as its capital, and not part of China under the definition of the PRC in Beijing.
Does this mean Taiwan is independent?
Yes and no. The government operating on Taiwan is a self-sustaining, fully functional, democratically-elected government unrelated to Beijing with its own economy, currency, etc. Whether or not Taiwan is an independent "country" is a very large grey area which cannot possibly be covered in the scope of this article. What's most important for visitors to understand is that visiting Taiwan from a legal and visa standpoint is that it is different from mainland China. Laws are different, visa regulations are different (click here to learn more about visa-free entry), customs regulations are different, and Taiwan uses its own currency, the New Taiwan Dollar (NT$, code TWD) and not the renminbi or Hong Kong Dollar.
Where can I learn more?
As any local will tell you, Chinese history and China as a nation has existed for over five thousand years. It is impossible to ignore the subtle cultural and historical influence upon the current political situation. If you really wish to understand the situation, try learning more about Taiwan island's history, and its development through each colonial rule (Dutch, Spanish, etc.) until the present. Also learn more about the Qing Dynasty, Chinese Civil War between the Kuomintang party (國民黨, KMT) and Communist party (共產黨, CCP), the Japanese occupation of various Chinese territories, and post World War II Asia (as in Japan's surrender and what happened to their territories) to get a more full picture of Taiwan. Although Taiwan was only discovered in 1544 by the Portuguese who named it Ilha Formosa (福爾摩沙), the island has gone through incredible social, economic, and cultural changes as it was handed from one power to the next in only a few short hundred years, not to mention the rich Aboriginal culture of the native people of Taiwan, much of which still remains today.
More articles on GuideToTaipei.com about Taiwan's history: