From the Help pull-down under Photos FAQ:
To remove a profile's Primary Photo (the one that appears as a thumbnail at the top) first edit that setting in the right column of the edit page for the photo.
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.
If you need to completely delete a photo immediately because it contains offensive or illegal content that violates our Terms of Service, e-mail email@example.com. This should only be done in emergency situations, e.g. with inappropriate photos of children.