You can modify which student fields appear in the directory listing by going to the Admin/Configuration tab and selecting from the options shown below. All these fields are searchable and sortable in the Student directory. Only users with Super Admin role can access this configuration.
The Custom Field provides some flexibility to define your own definition that suits your school's organization. You can define the name of the field and designate it as either free-form text or a directory selection. The Directory Selection option allows you to pick a person or group from the directory.
Once you save your configuration, you can edit the student records and fill in the custom field accordingly.