"user TSconfig" and "page TSconfig" are very flexible concepts for adding fine-grained configuration to the TYPO3 backend. It is a text-based configuration system where you assign values to keyword strings, using the TypoScript syntax. The TSconfig Reference describes in detail how this works and what can be done with it.
User TSconfig can be set for backend users and groups. Configuration set for backend groups is inherited by the user who is a member of those groups. The available options typically cover user settings like those found in the User Settings module, various backend tweaks (lock user to IP, may user clear caches?, etc.) and backend module configuration.
Page TSconfig can be set for each page in the page tree. Pages inherit configuration from parent pages. The available options typically cover backend module configuration, which means that modules related to pages can be configured for different behaviours in different branches of the tree.
It also includes configuration for the FormEngine (Forms to edit content in TYPO3) and the DataHandler (component that takes care of transforming and persisting data structures) behaviours. Again, the point is that the configuration is active for certain branches of the page tree which is useful in projects running many sites in the same page tree.