You will need three programs running on your home or office computer to be able to do this.
1) A server program like Apache. There are a number of server programs available, besides Apache.. They listen to the internet for requests for information, and then send out the requested information. It is possible to have them listen to their own computer and respond to requests. Most server programs are available as free downloads. Apache can be found here: http://www.apache.org/.
2) Mysql. Mysql actually runs as a form of server on the computer, but as a database server rather than a web server. Mysql is a free download from http://dev.mysql.com/downloads/mysql/5.1.html for the latest version. You will need to install this and configure it properly.
3) You will need Php. This is available from http://www.php.net/ as a free download.
Make sure when you download you are getting the proper version for your operating system.
Installing on your home computer can be tricky. I would suggest you patiently spend a lot of time reading through the documentation, and possibly looking at web sites that give trouble shooting advice before even starting. Hit your local bookstore, or your local library. There is a great deal of help out there.
An I option I have used was to pick up an old computer (pre-1 Ghz), and set it up as a server, connected to my home computer via a router, and blocked from internet access. If you can find an old computer and a cheap router this is well worth the investment. I installed Ubuntu Linux server version, which has Linux, Apache, Mysql, and Php (LAMP) packaged together. It is one easy, quick install for the lot. Ubuntu is available at: http://www.ubuntu.com/.
A word to the wise: The environment on your computer is fairly constant as far as operating system, IP address, browser, screen resolution, and a number of other factors, which will vary with your users, when the site is actually up and running. Test your scripts extensively and throw it some curves before allowing the public to have a shot at it. Even then expect to have problems that will need to be corrected fairly quickly when the public is allowed access. Even software architects with years of experience will tell you that no roll-out goes without problems.
Good Luck