If you really want to host it on your computer, one thing to keep in mind is for it to be viewable all the time, your computer must be powered on 24/7. So if that’s the case, you’re not looking for hosting, just a free subdomain. Domains cost money, but the people who own them can make subdomains, and they can let people use them for free if they like. This is exactly what DynDNS is for, since I also assume you have a dynmaic public IP, and DynDNS will keep track of it and make sure the subdomain always resolves to your server. I like the .gotdns.com one, its easy to remember but .blogsite.org would also be good. DynDNS owns gotdns.com, and blogsite.org, so they let people use the subdomains of it.
But first, do you have a router? If you do, that means you need DynDNS to point to your public IP, the router’s IP (you can find it out on whatismyip.com), and then you need to forward port 80 to your computer’s port 80. Port 80 is always used for websites, if you didn’t use that then the user would have to type in for example website.com:37. You probably also use DHCP, so first you have to give your computer a local IP that won’t change, outside of the range of address that DHCP assigns. Than you can forward port 80 to your computer’s local IP’s port 80.
If you don’t have a router, than the DynDNS needs to point to your computer’s public IP address, and make sure the firewall allows access to port 80.
