You will need the Full Write People permission to perform this action, at one or more of the person's campuses. It is campus-scoped, so a campus override can change it at one campus. Anyone with this permission also has Limited Read People and Limited Write People. See People for what these permissions cover, and Campuses and organizations in Version 8 for how campus access works.
Not everyone wants to be listed in the people directory. To exclude someone, locate the profile you want to modify. On the person's profile, select the header, noted as Listed or Unlisted, beside the person's name.
OR
Select the Actions button in the top right, and choose Update Privacy Settings.
Finally, mark the profile as Unlisted and save your changes.
Profiles can also be unlisted en masse by custom criteria via the directory settings. See How do I manage the church directory settings?
Tip: If your account has more than one organization, each organization sets its own directory access and listed rules, so a person can be listed in one organization's directory and not another's.