How do I register a domain name, but keep who I am secret?

Swedish company PRQ offers anonymous domain registration for $40/year.

They're the company who hosted The Pirate Bay and Wikileaks websites, and have a good track record of not interfering or giving out customer data unless what you're doing contravenes Swedish law, which makes domains and sites hosted with them somewhat resilient to take down orders.

UPDATE:

The only thing that you need to be aware of is that linking fake, anonymous, or masked information to whois records could waive your rights to the website and its content in a court of law, at least in the U.S. In effect, you are passing the rights to the site and its content to the anonymous or private registrar; they now own your site. From Wikipedia:

"In a trademark infringement case, a 2009 United States District Court ruling in U.S.A. held that, for domains with “private registration”, the privacy service is legally the “owner” of the domain. The privacy service acts as the “cyber-landlord of the Internet real estate”, and the domain is “licensed” to the customer of the privacy service."

This is why some domain privacy services, such as the one from Dynadot, now list your real first and last name at the top of the domain record and mask only your address, email address, and phone number. It ensures you're still offered some privacy while being recognised as the legal owner of the domain.

Hover gives you free anonymous WHOIS data when registering a domain name.

Domain name registrars will hand over your information without fighting for you at the slightest legal provocation. These services are meant to keep your data out of the WHOIS record to prevent spam and general harassment, not as a legal shield for your identity or to allow you to do legally questionable things behind a domain.

Or, you could registar with Dreamhost which offers the service for free. Of course, that means only hiding your personal info from the WHOIS listings. But you will still be reachable through redirected emails.

That's more than enough for most people though.