The only way to register and start using a domain name is to use the services of a domain name registrar. The domain name industry is regulated and overseen by ICANN, the organization that is responsible for certifying companies as domain name registrars. At one time there was only one domain name registrar - Network Solutions, Inc. - but today there are dozens of accredited registrars. Only a domain name registrar is permitted to access and modify the master database of domain names maintained by InterNIC. The master database contains the documentation on all of the domain names registered to date. InterNIC provides a list of accredited domain name registrars here.
If you come across a registrar that is not on the list it is most likely that the organization is acting as a reseller for one of the accredited registrars as only an accredited registrar has the ability to modify the database of domain names.
ICANN-accredited registrars have the authority to assign domain names for the TLDs of .com, .biz. .info, .name, .net, and .org. ICANN does not, however, specifically accredit registrars to provide registration services for country code TLDs.
The registrar you choose will determine the cost for registering a domain name, but in general expect to pay a yearly fee. Some registrars will offer discounts if you register for more than one year, but no registrar has the authority to sell more than a 10-year registration contract for any domain name. The registrar will ask you to provide some contact and technical information to have on record. This information is stored in a registry, which other computers can access to find information on your site. While it may seem like a privacy issue, it is best to be honest when supplying the contact information to a registrar. The registrar can cancel a contract and resell the domain name if it cannot contact an actual person to settle legal issues that may arise.
There are many laws that regulate the registration of domain names, such as registering a copyrighted name or registering a domain name for the sole purpose of blocking someone else from using it. Check with your domain name registrar for the laws that affect how you register your domain name.
If you want to create a domain name and are not sure if it has already been taken, you can check a whois server that keeps track of all the registered domain names.