Right, you only rent the right to use a domain name. Unless you own the TLD itself, like Google and Yahoo! do.
As for hosting from home, if you have a small website, with small amounts of traffic, a line faster then DSL (High Speed DSL at the least, Cable, T1, etc.) and no large files hosted on the server (anything larger then an MP3 would probably cause a clog if more then a few people try to download. We're talking a few kb/s if that.
Your best bet, if you meet those specs, is to get an old laptop your not using, or an old PC in a closet. Set up remote desktop on them, stick it somewhere safe, and don't touch it. If you need to work on your site, use FTP and Remote Desktop as needed. But the less you mess with the server on a home setup, the better. Any little thing can cause a lot of users being lost.
Key Words: Always On.