To remove the photo from the profile, click on the photo. This will open the Photo Edit page. On the left side of the page, where it says "People, Places & Things in the Photo," select "REMOVE" next to the appropriate name(s), then click "Save Changes to Photo Info" at the bottom of the page. When you save the changes, the photo will not appear on any profiles.
This disconnects the image from any profile, but it will still exist in the database for a short period of time so that it can be recovered. This way photos can't be maliciously or mistakenly deleted.