Taiwan’s real name is “Republic of China” and China’s real name is “People’s Republic of China”. Before the Chinese Communist Revolution they were all part of one nation. Taiwan managed to not go communist and maintains that they are the real China. Communist China maintains Taiwan is part of China and won’t do business with anyone who calls Taiwan “Republic of China” and won’t recognize their sovereignty. It’s sort of (but not really) like North and South Korea.

This is a two minute explanation, the full history and intricacies are far more interesting and subtle.