Depends on what you want to do, how much traffic () your site generates, and how much web design you want to do, really.
() the amount of data to/from the site - popular sites hit several (or hundreds) of GB a month, but small sites of text would have limited traffic.
As others have said, you need to register a domain name, and consider web hosting.
In some cases the registration firm offers 'basic' facilities that allow you to 'point' visitors to a site - so you might have a Facebook page, or Blog, or some other 'hosted' page(s) and can make 'www.yourdomain.com' point to the destination you want.
For .co.uk you could use Heart Internet which provides a small amount of webspace and low traffic limit (100 MB a month) so if you had loads of photos it may not be enough.
For .uk the managing organisation is Nominet, which allows domains to be registered for 2 years (and renewed for 2 years).
Cost is pretty low - expect to spend under 8 quid for 2 years. Go to Heart Internet, or 123-reg.co.uk, or ukreg.com as example firms (hundreds more, but no need to spend a bundle, and remember you can register a domain with one but not use their web hosting, so don't feel obliged if they charge a lot...
For other domains, prices are more varied (.com, .org, .net are usually around $10 a year, while .info is often cheaper {for the first year}) and obviously varydue to exchange rate. However, domains can be registered/renewed for multiple years, so some of mine have been registered for up to 10 years {when exchange rate was good for us}. To save cash, you can transfer a domain from one firm to another, as they often give a discount for transfers (to encourage people to switch and use hosting or other services).
Something to mention on registering domains - depending on what you plan to use the domain for, is that you might not want your postal address and phone number to be available. Clearly if you plan to run a business, then it would make sense to have the correct details visible (often called 'WHOIS' details).
Sometimes an organisation (or an individual) wants privacy, and for non-commercial sites, that's allowed for .uk but with the others such as .com etc, you can often opt for it whether running a business or not - sometimes it is free, often there is a fee.
If you decide to register some names, you might want to try
- GoDaddy.com (.info costs either $1.99 and sometimes less - register for a free account and they send discount offers quite frequently, eg 25% off .com, .net, .org at present)
- Name.com (offers free privacy, and you can build your website with some tools from Google, for free... .info costs about $4)
If you visit the Heart Internet site, click on Domains and you will see a list of what they include - hover and each item should be explained for you.
Good Luck!