Firstly, if you're setting up a new business, I'd suggest that you check that a business or limited company doesn't already exist with the name you'd like to use. If you set up a company with the same or a very similar name, customers may be confused or the existing company may take legal action to prevent you from trading under that name - not a good start!
The website start.biz is worth a check because it has a search for business names that are already being used - see http://www.start.biz/business_names/multi_search.php.
Once you've checked that no one else is using the same business name, check whether your domain name is available by doing a search on the 123-reg site - http://www.123-reg.co.uk/. Consider using hyphens in your name if your first choice isn't available, or to make the name clearer. For example, my wife's business Vanna Art uses http://www.vanna-art.co.uk/ because www.vannaart.co.uk runs the words together and is more cumbersome.
For a UK business or a personal website, the suffix .co.uk is normally the best option to use as it is widely known, good value for money and links you to the UK. (Cost at time of writing is £6.88 inc. VAT for two years). Non-profit organisations can use .org.uk or .org. The use of certain suffixes is restricted - e.g. ac.uk, which can only be used by higher and further education and research institutions.
There are also rules about the lengths of domain names:
.co.uk and .org.uk must be between 3 and 23 characters long, not including the .co.uk or .org.uk part.
See http://www.nominet.org.uk/registrants/aboutdomainnames/rules/ for full details of .uk web domain rules.The Business Link site also has useful information on choosing a web domain name for your business.
