Does Anyone Know What "Private Listing" In E-bay Means....

This doesn't mean as much as it used to since ebay doesn't list the full usernames of other bidders
in regular auctions anymore anyway. Private listings give a buyer more privacy but it makes it harder
for a buyer to see if there was any shill bidding going on.

I assume that the buyer knows that they bought the item sooner or later....

determines that? And how is it done?

If I wanted to assure that I remained somewhat anonymus when I bid - how do I do that?

See: http://pages.ebay.com/help/sell/private.html

If you want to be somewhat anonymous you can create additional accounts:
http://pages.ebay.com/help/account/questions/second-account.html

And you can make your feedback private:
http://pages.ebay.com/help/feedback/profile-public-private.html