There are 5 “reserved” groups that are automatically created based on a contact's activity:

Contacts can be part of multiple groups, and these groups cannot be deleted or edited.  

You can create additional groups  (ie: Board member, Volunteer, Major Donor, Offline Donor, etc) to help you segment your contacts further in your Community.

Step 1: Navigate to your Community and select "Create Group" in the group listing on the left.

Step 2: Write your group name and save.

Step 3: You can assign contacts to the group when you select the contact and add them to the contact card (shown in the sidebar).  

You can also create a new group by typing in a unique group name on the contact card directly.

You can edit or delete a custom group at anytime.