with the appropriate high level domain administration authority. In the case of names with a .uk suffix this is Nominet UK.
You can register a name with any one of hundreds of authorised registrars, and these usually tend to be hosting companies, so you can register your name via the company which will host it.
Once registered, your name must be 'tagged' with the IPSTAG relating to the nameserver of your host. This information will be propagated to the DNS databases around the world, enabling web browsers to locate the index page for any website that is published to the name when someone types its url into a browser address bar.