If you want to run a server at home, check with your ISP! Many prohibit running live web servers and actually block the ports so that you can't. If your ISP allows you to have a server, then setup one up. There are many FREE open source systems that can run on just about any OLD machine. Try Ubuntu Server, or Fedora, or Suse Linux. The server software includes Apache, MySql, Php and an email system!
Here is one good article to setup a Linux based web server with a "hosting" control panel to make the operation easy.
http://www.howtoforge.com/perfect-server-ubuntu-9.04-ispconfig-3
If your ISP doesn't allow servers you can still set one up for use inside your home!
Edit -
Well if your ISP allows that, just register a DOMAIN using any of the registry sites Domains410, TwoCows, GoDaddy etc Then setup the server the guide above is a Step By Step for you.
You will use YOUR domain name. You can also have the registrar act as the DNS for you in most cases, cost a few extra dollars but saves you having to have two IP's.