The Short Answer: A Domain Name Registrar is a company authorized to sell and register Domain Names. Once you buy a domain name, it is registered to you and you own it.
The Long Answer: Domain Name Registrars are accredited either by the ICANN (Internet Corporation for Assigned Names and Numbers) or by a national ccTLD (country code Top-Level Domains) authority to register Internet domain names.
ICANN has authority over gTLDs (generic Top-Level Domains). Examples of gTLDs include domain names ending in .com, .net, .org and .mobi.
ICANN does not have authority over ccTLDs, although it is quite common for Domain Name Registrars to offer ccTLD registration services along with their gTLD registration service.
For More Detail: visit the Wikipedia page on Domain Name Registrar. Great source of information here.