To dive more into Gizmo's explanation, getting a domain, not that expensive, just a pain to find one that you like and that is available from sites such as godaddy.com, or you can obtain a free domain name from a site like freedomain.co.nr/ you just have to be willing to put up with them tagging on an extra bit to your domain name.
In regards to the server, if you are going to just run a personal sort of website, nothing to big or fancy you can find plenty of web hosting sites out there, they give you a reasonable amount of space, email address, stuff like that for a low monthly fee.
The downside of this is you have no direct access to the server and limitations on what you can actually host there.
If you go with a VPS (Virtual Private Server) it costs some more money, but you get direct access to your VPS via ssh and can do more things with your site, these like web hosts limit the amount of processing power, space, and other resources that you can use.
If you go with a dedicated server, it costs a lot more money but you have full access to the server and can do anything you want with it (As long as it is within the hosts TOS) Its like owning your own server just having it located somewhere else with a faster network.
When deciding with option to go with you need to assess what your are going to do with your website, IE:
Personal Website (John's Home Page)
- With pics, blogs, small web board
A web host would be fine for this sort of plan
A web forum, larger site, advertising
- Containing downloadable file, large user hits
A VPS would be fine for this sort of plan as it has more HD space, a bit more memory, and a faster network
A larger site, IE: Torrent Site, Large Web Forum
- Containing a large amount of users, databases, files
A dedicated server would be better geared towards this setup, as you have full control over the server, full usage of the servers resources, and a fast network.
Unless you try something to which you have not already succeeded ~ Then you shall NEVER grow