Yes, you have to have your computer on for the website to be up. Your address on the Internet is the same as your IP address as well, unless you buy a domain name or get a subdomain set up with a DNS.
http://www.dyndns.org/ offers free subdomains with DNS (as opposed to a redirection service). They also have clients you can install that automatically update their DNS server every time your IP address changes. It's also free if you just get a subdomain. If you own a domain name you can use that as well.
Gamer1:
30KB/s eh? Yeah, that's twice my upload speed.
Anyway, the 5 tips and stuff are just for people who haven't tried this stuff before.
I never much liked the idea of updating something manually whenever my IP address changes. I prefer having an actual name attached to my IP via DNS over redirection anyway, so cjb.net is out for me.
I have a Pentium 233Mhz PC with 64MB SDRAM and a 2GB HD with Debian Linux set up as a webserver (Apache with PHP and MySQL), a mail server, and an ftp server (real users only). So far it seems to be working with dyndns.org, but I only tried setting all of this up tonight. Everything's secured down, and the only ports available (redirected from my router) are the absolute minimum needed (21, 25, 80).
I think I'll install squirrelmail on it so I can easily check the mail on this machine from my other computer.