Every computer on the Internet has a unique address, this is a collection of numbers, which is not the easiest of things to remember.
To help locate the particular web site you want, out of the millions out there, the Domain Name System (or DNS) links more memorable names to these numerical addresses.
When you type www.zuzzinternet.uk into your web browser, your device automatically uses the DNS to find the address of our server.
Domain names are generally used to locate web sites and email services.
To be a bit more pedantic, in the example above, it is only zuzzinternet.uk that is the domain name, as a domain name can point to more than one computer, the www prefix is used to say that the required server is the one providing the web site, equally, the mail prefix (as in mail.zuzzinternet.uk) would point to the server providing the email services. The www and the mail prefixes may both point to the same server or they may point to different machines.
Incidentally most browsers will automatically insert the www. for you so typing zuzzinternet.uk will have the same effect as www.zuzzinternet.uk
Domain names are made up of two parts; the domain and the Top Level Domain (or TLD), in our case, zuzzinternet is the domain and .uk is the TLD.
TLDs are organised by country .uk for the United Kingdom, .fr for France and so on. As the creation of domain names started in the USA, the TLDs of .com, .net, .org etc are actually for American addresses, however they have come to be regarded as International for organisations who do not wish to be country specific.
Within a TLD such as .uk, there is a sub-category of address such as .co.uk, .org.uk, etc. Recently it have been possible to register a TLD of just .uk which makes life easier.
If you want a domain name, you can register one through a recognised Registrar (like us), you do not own a domain name, but rent it for a period of time.