You can buy (register) a domain name on some of the popular domain registrar websites listed below (the list is suggested by the discussions on Quora and is not exhaustive):

• namecheap.com

• godaddy.com

• domain.com

• iwantmyname.com

• netlynx.com
  

• gandi.net

The domain name will be billed every year, and the yearly cost of a normal domain name ranges from $1.99 to $29.99 a year, depending on the domain extension such as .com, .net, .biz, .online, .xyz etc

When looking for a domain registrar, you should keep in mind:

• Price
  The price you pay for your domain can vary greatly between registrars. When comparing prices, take into account the other services included by the registrar, their customer support, testimonies, etc.

• Accreditation
  Your chosen registrar should be ICANN accredited. ICANN maintains a list of accredited domain registrars. If the domain you have chosen is a reseller of an accredited registrar, they should let you know which registrar they represent.

• Terms
  Take the time to read and understand your domain registration terms. Pay attention to ownership clause and other terms of condition.

• Duration
  Do not register your domain for more than one year in the beginning. That give you time to evaluation your registrar before further commitment, as well, allowing you time to evaluate your domain name in case you wish to use a different domain name.

• Delay
  A domain name registration generally takes no more than 48 hours.

• Transfer Policy
  You should be able to transfer your domain to another registrar without any restriction other than those enforced by ICANN. Check the registrar for other restrictions and curbs.

• Customer Support
  Check if phone, chat or email support is available.

• Management
  Does the registrar provide a domain name management panel for you to manage your own domain name?  Is it easy to make changes on the DNS yourself? Or do all changes have to go through them?