It is very easy to check the availability of and domain name (website name), simply use a domain name registrar. There are thousands of these online and you could use anyone of them to check or register your desired domain name but you will be better off using one of the bigger ones like:
- 123-reg.co.uk (the most popular registrar in the UK) or
- godaddy.com (probably the most popular one in the world).
Go to either of these websites and you will find a box where you can type the website name you are interested in together with the extension (.com or .co.uk etc.) you require. Then click the search/go button.
The website will then check to see if the website name with it’s the desired extension is available or if someone has already registered. If it is available you will be able to buy it there and then. If it’s not you will probably be able to get it with a different extension or alter it by adding suitable prefixes (such as: my… or the… ) or suffixes and searching again until you find one you are happy with.