Before you install a chat room on your web server, you need to ensure that your web hosting company provides the right kind of platform.
Almost all paid hosting plans provide what you need. For example, Website Palace's hosting plan comes with a pre-installed chat room script that you can install with the click of a button (free web hosts do not provide this.)
So all you would do is login to your hosting account, go to the area where the free applications are located, and instruct your host to install the script and create your chat room.
Your chat room will be up and running in no time.
Gone are the days where you have to download a script, configure, upload and launch. Hosting has come a long way and installing a chat script is no longer the headache it used to be.