"Chinese Taipei" is a designation often used by Taiwan when participating in international organizations. The name is used at all sporting events, including the Olympics, the Asian Games, and the FIFA World Cup.
Even though Taipei is only a city in Taiwan, the name refers to the entirety of Taiwan as well as the minor islands of Penghu, Kinmen, and Matsu.
The peculiar name came about due to conflicting attempts by Taiwan and Mainland China to compete in the Olympics under the name China. That's because the government in Taiwan is called the Republic of China, while the government in Mainland China is called the People's Republic of China. By 1979, many countries and the UN had recognized the government in Beijing as the only authority in all of China.
The name "Chinese Taipei" was the compromise that allowed both sides to continue to participate in the Olympic Games. The vague description of "Taipei" allows the government in Taiwan flexibility in describing its national boundaries, while "Chinese" relates to the country's official name.
But the name is not without controversy. While Taiwan intended the word "Chinese" in the title to mean Zhonghua, or Chinese civilization, Mainland China translated "Chinese Taipei" as Zhongguo Taibei, meaning "Taipei, China". Mainland China eventually changed its translation to be consistent with that of Taiwan's ahead of the 2008 Beijing Summer Olympics.
Taiwan also goes by the name "Chinese Taipei" in the Asia-Pacific Economic Cooperation (APEC) and when it participates as an observer in the World Health Organization, the only UN agency in which it is allowed participation.
But "Chinese Taipei" is not the only name compromise Taiwan uses internationally. In the World Trade Organization it is referred to as the "Separate Customs Territory of Taiwan, Penghu, Kinmen, and Matsu".