Public Groups is one of the most famous and widely use feature in Salesforce, with contain a set of users, it can be individual users, other groups, the users in a particular role or territory, or the users in a role or territory plus all of the users below that role or territory in the hierarchy.
Only admin (or users with Managed Users permission) can create Public Group. Public Group can be used from sharing rules, manual record sharing, report/dashboard folder sharing, list view visibility and etc.
What is Personal Group?
Personal group is not so popular compare to Public Group. Personal Group are intended only for the user themselves to create and manually add members.
Salesforce user can create and manage their own personal group from Name | My Settings | Personal | My Groups, from here you click New Group or edit or delete existing group. Each user can create more than 1 personal groups.
While Public Group only for system admin to setup, Personal Group can only be created, edit, "seen" and used by the User himself (no one else can access it).
Personal Group Usage
Once your personal group created, you can use it to share records to this specific group. But, make sure your Salesforce admin to add Sharing button to the page layout.
If there is no personal group created, you will not see Personal Groups option in drop down menu. Before Personal Groups created:
Once Personal Groups created: