Well if you have a broadband connection at home, what i think is your case, you have a 'real' ip address that means other computers connected to the internet can access your machine remotely. This is good and bad.
If you decide to run your website from your machine you can. You just need to install a good firewall, iptables would work fine, a good webserver, httpd is the best bet, keep your home pc online all day long and find a good and free redirection service. I am not sure if that is the name but basicaclly they would translate the user's input, say yourwewbsite.com, to a ip address that happens to be your home server, say 188.8.131.52.
Or you can just pay a fee and have some isp do that for you.
I would go with the first option. Good luck.