Domain names have to be obtained from registrars: companies accredited by ICANN, the Internet Corporation for Assigned Names and Numbers.
You can register the domain yourself or get someone else (e.g. a web designer) to do it for you.
I generally use Hostinger to register names, because I also use them to host my websites. You can check domain name ideas and availability using Hostinger’s domain checker.
If your ideal name has already been taken, I’m happy to explore other options with you.
Note that prices vary between different top level domains (TLDs; such as .com, .co.uk, .online) and also that special offers are usually for the first year only (make sure you check the renewal price before committing yourself; £5 for a fancy-sounding TLD might seem like a bargain for 12 months, but £30 every year after that might not!).
Although I usually register domain names for my clients and create their websites, I do not consider either the names or sites to be my property, so you will own your domain name and your website.
In the case of the name, it is usually more convenient for me to register it – and to pay the relevant fee – and there is the advantage (as some see it) of the client’s name not appearing on the Whois register (something that can also be avoided by paying for privacy protection).
If I register it and you want to take your business elsewhere at some point, then I won’t charge for handing it over; if you’d rather register the domain name yourself, that’s fine with me (registering it via Hostinger, although not essential, will make life a bit easier when setting up the site).