it might be an idea to talk to your ISP about this. Many providers will not allow you to run a server on port 80, and that means nobody will see your server, even if they have your IP address.
On the subject of IP addresses - do you have a static IP or does your ISP assign one dynamically? Do you already have a primary and secondary DNS server? You'll need both if you want to be sure that your site will have the maximum amount of up time. Once all that's sorted out you'll need to look at firewalls and routers, server Operating systems and which ports to lock down to prevent the nasties out there from compromising your server.
Once you have all the boxes ticked, so to speak you can run pretty well any PC as a web server, provided you're prepared to leave it running 24 hours a day, seven days a week, connected to an ADSL line. Many people are under the impression that a web server has to be an expensive monster with tons of RAM and a state of the art processor, but that isn't the case. Most modern PCs will function very well as servers, although if your site starts attracting anything serious by way of traffic you'll need to upgrade your hardware pretty smartly.
Frnakly I can't see the point of running a server at home when for a relatively small outlay you can run a dedicated rack server in a professionally monitored server facility. All your needs are catered for, and you can even administer the server from the comfort of your home PC.