Private Groups -- private collaboration among members without a public presence on the Zotero website. Only users who are members of a Group are able to see the Group's site.
Public with Closed Membership -- Groups a closed-group environment with a public presence. This provides the group a method of sharing their bibliographies publically with other Zotero users, but joining the group remains invitation only.
Public with Open Membership -- Groups are open to other Zotero users and anyone can join the group.
For more on group collaboration click: http://www.zotero.org/support/groups