Notifications let users when there are updates in the file system that require their attention. The goal is to limit the number of notifications served up to the user and have every notification served progress the user further (as opposed to being purely informational with no action).
Types of updates notifications handles:
Notifications are viewable in the finder and browser navigation bar. Notifications also pop up in the upper right hand corner of the browser when they come in with ability to action on them. Users can turn off popup notifications for certain types of notifications or completely disable it.
Within Finder, notifications live in the the sidebar. They share the same UI as notifications in the browser.