Taiwan IS the Republic of China, it became the home of the government in exile after the Socialist Revolution, the ROC still says they are the true government of all of China which is where the whole "One China" Policy comes from. Meanwhile Mainland china says they control Taiwan.
The Peoples Republic of China and the Republic of China do not recognize each other as legitimate governments. The case for the PRC is pretty solid though and they are the true legitimate government. Meanwhile Taiwan is not recognized as such by many governments but likely would be if they dropped their claim to mainland China. China could have easily forced their hand but they are pretty sane and realize that doing so would be tantamount to suicide.
China is not a threat to us from a military standpoint, they are completely beholden to the rest of the world and woudl have far to much to lose. They make the things we want, the computers and iphones and tv's you are currently looking at, etc. It is not like the Soviet Union where they make nothing we want.
The combination of Socialism and Capitalism they have is pretty amazing and the quality of life of Chinese citizens has moved so far ahead over the last 40 years it is unreal. The average Chinese person on the street loves America.