It's a bit complicated, but here's the gist:

Look on a map and you see a big country called China. Originally this was called the Republic of China (ROC) but there was a Civil War between the ROC and the new Communist Party of China. The ROC eventually lost and fled/exiled to the island of Taiwan. The Communist Party took control of China and renamed the country to the People's Republic of China (PRC).

The ROC view themselves as the legitmate government of China as they never "left" (the island of Taiwan was controlled by the ROC). The PRC obviously disagree and in their eyes the island of Taiwan belongs to them as it was a part of China and they are now the government of China.

For various reasons, the PRC is/was unable to take control of the island militarily. With the opening up of China to foreign countries/companies/etc the PRC has mandated a strict "one country" system whereby the international community can not recognize the government of Taiwan (which is still called the ROC) as an independent state but instead a separate "system" within the PRC. The "Chinese Taipei" thing is one of the ways that the international community can recognize Taiwan the people/cities/place but not as a country in its own.