Permission groups allow you to create and manage groups of staff members with the same permissions. You can think of them as different roles that staff members fulfill. Users do not have individual permission settings, user's effective permissions are the sum of all of the permissions granted by the groups they are a member of.
To create a group, use the Permission Groups section of the Configuration tab.
When creating or modifying a group you can select which permissions you want to give it.
The "manage staff" permission allows users to manage the permission groups. You can only manage groups that grant a subset of your effective permissions. You cannot assign permissions you do not currently hold. This is a security precaution that prevents users from escalating their permissions beyond what was explicitly granted.