As the others have said, Taiwan is an independant country that is in Asia and is considered an economic power house. A lot of computer parts are made in Taiwan or by a Taiwanese company (ie. Asus, Acer, BenQ, Leadtek, MSI, Gigabyte and many more). Taiwan's status is only not defined because of PRC' and mainland Chinese' irgnorance.
Same culture, same language/dialet means shit. Countries(independant political entity) are not bounded by such rubbish. Take a look at South Korea and North Korea, they eat the same food, share the same culture, blood-related etc.
Taiwan is actually the name of the island. The country's name is Republic of China(ROC). ROC's seat in the UN was replaced by the People's Republic of China in the early 1970s, however ROC was offered another seat but Chiang-Kai-Shek forfeited the seat because he insisted "one or the other".
The Taiwanese people have been separated from the mainland for 50+ years, their dialect has changed, the language has changed. Taiwan uses traditional Chinese while the mainland uses simplified Chinese.
"I am Taiwanese"
"Taiwan is an independant nation"