To add to Chuck's answer, you could buy a domain name so long as it hadn't already been registered. If you fancy <your surname>.com, you're almost certainly out of luck unless you have an unbelievably rare surname, as all have already been registered many years ago...
There are many sites where you can check if a domain is available or not. This is a good one:
http://www.hostinguk.net/domain_names.asp because it will not only show the domain you searched for, but will also allow you to look for others with different prefixes. E.g. supposing you wanted to register rae.com, it would tell you that it had already been registered (not by me, I hasten to add!), but would then allow you to check for rae.co.uk, rae.biz, rae.net etc...