EDIT
I misunderstood the question. I'll keep my old answer up incase it helps anybody :)
To install go to Add/Remove programs (or Programs and Features in Vista) and on the left bar there will be "Add or Remove Windows Components" in XP and "Turn Windows features on or off" in Vista (WHY WITH THE NAME CHANGES DAMN-IT!!!??) Find "Internet Information Services" and check it if it's not already. Go ahead and install it.
After installation, go to Administrative Services then to IIS Manager. Here you can configure your IIS installation.
I'll play around with it myself and post back anything I find out. But I'm not an ASP.net guy. I like PHP :)
Old Answer
Note if you didn't already know: if you set up a webserver on your box all you have to do to get to it from within your network is go to http://(your server's IP)
To find out what IP you have in windows, Hit Start then Run, type cmd hit enter, then type ipconfig and you will get something like:
Windows IP Configuration
Ethernet adapter Local Area Connection:
Connection-specific DNS Suffix . : <BLABLABLA>
IP Address. . . . . . . . . . . . : 192.168.1.155
Subnet Mask . . . . . . . . . . . : 255.255.255.0
Default Gateway . . . . . . . . . : 192.168.1.18
Ethernet adapter Wireless Network Connection:
Media State . . . . . . . . . . . : Media disconnected
The "IP Address" (192.168.1.155 in my example) is your local (i.e. only valid from behind your router) IP
The easiest way to do this would be to install a web server on your old box then set up your router to forward port 80 to that box (assuming you have a cable connection). Look at http://www.portforward.com/ for details on that.
Then you can use your router's IP address to access your websites from outside of your network. To find that IP address go to http://www.whatismyip.com/ from inside your network.
So, for example (fictitious IP warning) if you IP were 300.424.234.123 then you could get to it from outside your network (anywhere on the internet) by going to: http://300.424.234.123
If you want to serve a domain things become trickier. You would have to register the domain with a company like http://godaddy.com then point it to your IP. You would then have to configure your webserver to serve the correct documents depending on what the requested domain was. In Apache you do this with VirtualHosts. I have no clue on how to do this with IIS.
Those are the steps involved. If you have any questions about a particular one I would be happy to help :)