If you have a website, you obviously need a domain name. There are many choices for registering a domain name. Keep in mind that a domain name registrar must be accredited by a generic top-level domain (gTLD) registry and/or a country code top-level domain (ccTLD) registry. Domain registry services range from very cheap to very expensive. If you are in doubt, check ICANN-Accredited Registrars.
To register a domain name, you will need to go to a registrar and perform a search to see if it's available. If it isn't, try modifying the name you would like and check its availability. If it's available, purchase the domain. Often you will be offered the option of paying a number of years in advance, at a discount. Paying the registration fee gives you the right to use the name for the specified time purchased. When the term of use expires, you will need to renew the registration. Usually the fees don't change.
We advise registering your domain name instead of having a web developer or hosting company register it, because it will be in your name, you will have control without relying on a third party to keep it registered. And if you decide to change your web hosting you may move your files to a new host and point the corresponding domain name to the new web host.
There are companies who will purchase the domain name for their clients but show themselves as the owner and if the client moves on to another service, only release the domain name for an additional fee (whatever value they place on the name ... e.g. $2500+). You might also choose to transfer registration from one registrar to another which also comes with a fee.
It has always been our policy that we don't register clients' domain names and hold the registration in our name. But we are available to help with the process.