What was the significance of taiwan after the red army took control of china in 1949?

Taiwan became important because an exile government of China was formed there that claimed to be the rightful government of China. This caused China to have two governments, one that ruled China which was the communist one and one in exile which claimed their right to have power over China but couldn't actually use that power in any way. The problem still exists today and other countries have mostly been neutral on this question.