Notifications are for administrators in the Groups application. Read more about administrators here. They are a way to keep in touch with what's happening in your church's groups and be alerted to a potential problem. When/why notifications are created is really up to you.
There are two basic types of notifications in Groups: alerts and applications.
Alerts help you, the administrator, know if there's a potential issue within a group. Currently, the only type of alert generated by Planning Center Groups has to do with member capacity. In the group settings you'll find an option on the members tab: "Notify admin if enrollment exceeds __." Using this setting, you can set a threshold for the number of members that should be in a group, without auto-closing the group. For example, if small groups are really supposed to be around 12 people, you can set an alert to trigger if membership exceeds that total. Group leaders are not made aware of these notifications, unless they also happen to be administrators of Planning Center Groups as well.
NOTE: This is an area of Groups we're still developing as we get more feedback from customers. We started with only a few simple notification types and will expand them as people find more needs. Let us know what types of notifications would be useful to you by contacting us in the helpdesk (click the "?" in the top right of the application).
Membership requests are the second type of notification. They give you, an administrator, some insight into which groups have membership requests that haven't been replied to by their group leader. From here, you can also approve or reject those requests (if the group leader is out of town or isn't good about getting back to people asking to join the group).