Domain privacy is a service offered by a number of domain name registrars. A user buys privacy from the company, who in turn replaces the user's info in the WHOIS with the info of a forwarding service (for email and sometimes postal mail, done by a proxy server).
Level of anonymity
Personal information is typically collected by these registrars to provide the service. Some registrars take little persuasion to release so-called 'private' information to the world, requiring only a phone request or cease and desist letter.
Others, however, treat privacy more seriously, and host domain names offshore, even using e-gold or money orders in transactions so that the registrar has no knowledge of the domain name owner's personal information in the first place (which would otherwise be transmitted along with credit card transactions). It is debatable whether or not this practice is at odds with the domain registration requirement of the Internet Corporation for Assigned Names and Numbers (ICANN).