When you go to register your domain name more than likely you will be offered a plethora of additional services.  This may include: hosting, e-mail accounts, web page design, etc.  One of these will no doubt be privacy for your Whois information.   Whois is a search tool that allows a person to see who owns and is responsible for a domain name.  Basically it will give you the registrant, administrative and technical contact for a domain name.  In fact if you use Domain Tools Whois tool you can even find out where it's hosted, how long it's been and how many other domains are owned by the owner.

If you don't want your information out there for anyone to see you can purchase domain privacy.  This will replace your Whois information with proxy service contact information.  One company you may be familiar with is Domains by Proxy.  This company is owned by Godaddy and they usually will provide a free years service when you first purchase domain names, however after a year the price rises to about $7 or so.  To stop the service you need to logon to your Domains by Proxy account and un check the service for each domain and wait a few days.  Sometimes this doesn't work and you have to call up to Godaddy to get it fixed.  It can take some time so don't do it just because it's free on the other hand it can be very useful.

Should you get domain privacy?  Well like everything in life it depends...  If your buying and selling domain names you probably don't want it.  First some people will use their Whois information for advertising that a domain name is available by editing the contact information to include "Domain Name for Sale".  Second if someone researches your domain name and is interested in contacting you to buy the domain name they are more likely to contact you if they have a real person and real e-mail to contact in the Whois information. 

On the other hand if you only have a few domains and your planning on just developing them for yourself or your company it might be a good idea.  First your contact information may be "scraped" gathered by people compiling lists of contact information.  Although this doesn't happen as much with Whois information as it did since they have implemented a further step to access the contact information.  Second it will screen all the "phony" or low end offers to buy your domain name.  If people have to work to get your information they are more likely to be serious buyers, but you may not even want any buyers.