There are two things you pay for when setting up a website: the file server to host the site -- where you store the files that make up your web site -- and the domain. Let's start with the domain.
What *is* a domain?
You've probably already heard or seen someone mention "URL"s. That stands for Uniform Resource Locator, which is a fancy way of saying the directions to stuff you want to see on the 'net. URLs are a subset of URIs, which is where you start needing a degree (or matching experience) to keep from getting lost in the techno-speaking. Domains come in three parts: top-level, site, and sub-site domains. The first, top-level domains, are your .coms and .nets and all. The next up, the site domain, is the part that goes yoursitename in yoursitename.com. Sub domains include the www. or blog. or whatever. All together, they look like: sub-domain.site-domain.top-level-domain.
CAUTION: Each site domain is specific to a top level domain (TLD). Yoursite.com is not the same registration as yoursite.net or .org or any of the other TLDs. If you want yoursite.com and .net and .org and any other TLD (or want to prevent someone else from registering variants on "yoursite"), you have to register for each TLD separately. Most brand concious businesses stick to registering the .com, .net, and .org versions of their site.
Where do you get them, and what do they cost?
Something to make clear, you cannot "buy" a domain--you only lease it. The cost depends on who you register through. You can find accredited domain registrars, entities authorized by ICANN (a.k.a. the guys who run the internet) to lease out domains, through InterNIC. A number of site hosts will also provide domain registration services, though these are usually through reseller agreements. All in all, expect to pay from US$8 to US$15 per year to lease a domain under the .com, .net, and .org TLDs.
You can register your domain lease separate from your hosting service, or you can pick a package. For the most part (i.e. read the fine print on any package deals), your domain registration is separate from your site hosting service.
CAUTION: There's a recent move toward "proxy registrations". One of the pushes for it is that the information you provide when registering for a domain is posted to the WHOIS database as required by existing laws, which privacy advocates find a cause for concern. With a proxy registration, a proxy company registers the domain and sub-leases it to you. I do *not* recommend proxy registrations.
What style of domain do do you want?
AuthorsFullName.com is a pretty popular site domain format - there's laurellkhamilton.org and stephenking.com and ericflint.net and davidweber.net and ... well, a lot more. Sometimes, though, something in our stories will suggest itself. Patricia Brigg's hurog.com is a fair example of that. Piers Anthony, on the other hand, went for the friendly hipiers.com, and let's not leave out our best selling author, John Foxjohn at johnfoxjohnhome.com.
What you decide is ultimately up to you. Have fun with it. =)