omg!
lelliott, it is definitely better to have a server on your computer if you can do it. and you know why? becase it's freeeeeee!!!!
here's what you need to do:
1. set up Apache and other software (MySQL, PHP? - whatever you need) to work with the Internet. you can test your server simply by connecting to your IP address.
2. register a domain. you can get one for $8.95 at GoDaddy.com. also I have seen a company selling them for $7.95, but i don't remember their name.
3. go do zoneedit.com or granitecanyon.com and get a FREE dns service. (Zoneedit is much simple, but you will have to pay if you get more than 5 domains.) they have good instructions as to how to set everything up. their service will make sure that
www.yourdomain.com points to your computer even if your IP address changes (!!!).
that should be it!
of course you might want to set up a mail server or something like that, but that's a different story... good luck!