A domain is the address by which an internet browser is directed to the server’s location (which without a domain is just a string of numbers called an IP address, ie 218.439.36.99). Domains are registered at what is appropriately called a “domain registrar”. Sometimes this process is complicated by hosting companies offering free or discounted domain registration services with their hosting packages, or vice versa, domain registrars offering hosting services.
I recommend that domains and hosting should be handled separately, as this maintains a certain level of flexibility down the road should you decide to change either hosting companies or domain registrars. Having them tied together can complicate this process. The pricing of a domain is low enough that for anyone serious about starting a Web site, a “free domain” really is not worth the potential headaches the “freebie” can bring.
Domain pricing vary widely based on the Top Level Domain TLD (.com, .co.uk, .net, etc.) chosen. For common TLDs, prices are generally start at about £6.00 per year with .com likely costing more.