Webserver

Of course you remember it from your college days. You needed it to set up your Napster.
But I digress.
(Read number 5 first.)

First and foremost: Weapon of Choice.
Pick a server. If you have Linux experience, I recommend a Linux server running Apache. If not, Windows works fine too. With Windows, you can go with the built in IIS or XAMPP which is an Apache distro packaged with PHP and MySQL (and damn near wipes your ass for you - I'm not joking).

Second: Static IP.
Your server needs a static IP on the private network i.e. 192.168.0.yournumberhere. Actually, the 0 in there could be a 1 or a 2 or anything for that matter. What ever the case, that last number has to stay the same. Your router may even have the option to reserve that number for that machine if you choose to stick with DHCP.

Third: Port forwarding.
When your router receives a port 80 request from the WAN, it needs to know where to stick it (pun intended.) Check out http://www.portforward.com/english/applications/port_forwarding/HTTP/HTTPindex.htm and find your router. It'll step you through the rest.
Check out http://whatismyip.org . It'll give you your public IP. Open that address in your browser and see if you can see yourself. Congratulations, ur on teh interweb. You're also not done yet.

Fourth: Names, not numbers.
This is a fun one. You need a domain name and you need to register it. How you do this is your deal. You can go pro with something like GoDaddy.com or NetworkSolutions.com (I believe these may need a few hours to propagate the DNS before you can use them - hey, the Internet is a big place). If you're not picky (or a cheapskate), you can pick a shared host like http://www.afraid.org . They'll set you up a free sub-domain account. Point it to your IP address and you're good to go.

Fifth: This is a bad idea.
No seriously. Is this a computer you are comfortable having on all the time and will you be using it for more than just a web server? If yes, then proceed by all means. Otherwise, get a host and forget these silly dreams of being your own host. My reasoning? Your machine eats power. It eats LOTS of power. You will pay more for the electricity than to have your server hosted for USD$100 a year. Possibly by a magnitude of 5 or higher. Not to mention, courtesy of a little something called NEXT or Near End Cross Talk, your upload speed most likely sucks. So make the call. Hell yes, it's fun, but it might not be smart.

Good luck, friend. You'll need it.
-The Altruist