A Domain Name is a unique 'User-Friendly' character based address. The Internet works using numbers (xxx.xxx.xxx.xxx), referred to as an IP (Internet Protocol) address and rather than having to remember a whole string of numbers each time you want to browse to a website, a 'name' can be associated to a given string of numbers.
In order to have your own unique website address, you must first register / purchase your domain name, after checking that its available. This is all carried out through a Registrar.
The Domain Name is not just for website addresses, it can also be used for email, e.g. firstname.lastname@example.org.
Once you are the registered owner of a Domain Name, referred to as the Registrant in official documentation, you are entitled to use it for the time it is registered. You specify how long you wish to register the Domain Name for at the time of purchase.
When the Domain Name is about to expire, you are given the opportunity to renew it for another period of time and continue using it. If you choose not to renew it, the Domain Name is removed from the Registry and becomes available for someone else to purchase.