Taiwan is officially the "Republic of China" (ROC) which is not to be confused with the much larger "People's Republic of China" (PRC). Unlike its much larger cousin, Taiwan is actually a republic in more than name, having undergone democratic reforms starting in the 1980s.
However, the country's status is contentious, with the PRC claiming the ROC government is illegitimate and referring to it as the Taiwan Authority. A lot of the international community, including the United Nations, seems to side with the PRC.
Taiwan on Wikipedia