Agreed it definetly sounds like you are after web hosting as well as a domain name. We use a company called 123-reg.co.uk. They are not the cheapest as we've found that the less you pay, the worse the quality of service. I also know of another company called warp hosts who are a small independent (mention AnsteyDesign if you use them). Finally, as it sounds like you're just after a deal, I've carried out some research and the top provider is a company called justhosts. Although they don't offer 'free' hosting they do offer free domains and very cheap hosting.
Just something to consider before choosing a host. It's important that you use a provider who is in the same country as you, otherwise your website could be hosted in, for example, America. Not much good if you are trying to capture an English market. Also, putting it as simply as possible, by going with a cheap provider you may find that your provider deals with dodgy sites - this can have a big impact on how the search engines optimise your site so best to do some background checks first.