Long story short, mainland China (PRC) and Taiwan (ROC) don't recognize each other so they don't want to stamp each others passports / stick visa in there. The pragmatical workaround in place right now is that both nations issue another document to visitors from the other country (i.e. like a visa, but on its own piece of paper in my overly simplified words).
The details are nicely described here for PRC citizens travelling to ROC and here the other way round. The following is a brief and less accurate summary of mine:
- From Taiwan to mainland this is called Mainland Travel Permit for Taiwan Residents.
Taiwanese can apply for the permit in China Travel Service (CTS) Limited's Hong Kong or Macau branches or through travel agencies in Taiwan.
- From mainland China to Taiwan you need an Exit & Entry Permit for the Taiwan Area of the Republic of China,
a document issued by the National Immigration Agency of the Republic of China