It appears that Google is testing out a new feature for Google+ users that will help keep them in touch with their most important friends. A new feature “Get Notifications” will help users “Never miss another post … get notifications whenever important people in your life share something new on Google+.”
The above screen appears for many users when they first log-in to Google+, then the functionality mysteriously disappears afterwards. Seeing that the notifications are simply for users already in circles, it appears the functionality to operate much like Facebook’s “Close Friends.” When a user who has been signed up to “get notifications” posts an update, the Google+ user who added them will receive an update.
Notifications will be for posts only, so comments or +1′s won’t trigger a notification.
To see the page for your self, simply visit: https://plus.google.com/up/subscribe
UPDATE: Google has officially announced this feature.