Originally there was only one registrar who for a period of 5 years was given an absolute monopoly on domain registration by the United States Department of Commerce. That registrar was Network Solutions. The top level domain registry had to exist as a monopoly in the beginning in order to guarantee that domain registration and domain resolution service was centralized and managed. This made many of the next developments possible, such as the World Wide Web (which couldn't exist without the Internet and a working DNS system). However, it became clear that Network Solutions was not providing adequate service and could not keep up with demand. The Shared Registry Service was created to solve these problems.
ACCREDITATION
As far as we can tell, becoming an accredited registrar is as simple as paying ICANN some money. We presume that having some sort of proof of the capability to serve DNS answers and maintain a database of registered domains is also required, but we cannot tell if ICANN actually tests the registrar's systems or performs any type of validation of capabilities or service.
An accredited Domain Name Service Registrar is a company that will get your chosen domain name inserted into the DNS system so that it will resolve properly (provided noone else owns it already). These companies charge a fee for this service which varies according to the top level domain in which the domain name resides and the registrar's price schedule. The reason you must register is so that information about your domain can be inserted in a central database at the root servers where everyone else on the Internet can look to find the servers that serve your domain.
Remember, all DNS servers are built to look for information at a special set of IP addresses and servers at InterNIC. Because the vast majority of networking software is also configured for these servers and cannot be easilly changed, you must register and use an approved top-level domain. Fortunately today you have several choices for your top level domain.
APPROVED REGISTRARS
There is an organization called the Internet Corporation for Assigned Numbers and Names (ICANN) who approves companies who wish to provide domain name registration services.
ICANN's website contains a list of approved registrars. DO NOT REGISTER A DOMAIN WITH A REGISTRAR UNLESS THEY ARE ON ICANN'S ACCREDITED REGISTRAR LIST OR THEY ARE PART OF THE GOVERNMENT THAT CONTROLS THAT TOP LEVEL DOMAIN (such as Italy's root NIC for .it ccTLD and France's root NIC for the .fr ccTLD ). YOUR DOMAIN WILL NOT BE PART OF THE DNS SYSTEM IF YOU CHOOSE A NON-ACCREDITED REGISTRAR! ONLY ICANN ACCREDITED REGISTRARS MAY MODIFY RECORDS AT THE ROOT NIC!!
ALTERNATIVE REGISTRARS
If you are required to download special software (other than BIND) to make your domain work properly on user's machines, you are NOT using Internet standard domains. All current software and DNS servers come pre-configured to support DNS at InterNIC.
If you use an 'alternative domain' nearly 99.9% of Internet users will be unable to reach your domain, website or mail server in your 'alternative domain'. Check the 'top level domains' page for a list of valid top level domains.