WARNING: This is not a general purpose description of how to set up a Ruby on Rails production environment. If you want to host a public Rails website, I highly recommend using Apache with Passenger, which is very easy to install and maintain.
From your description, it sounds like you are working with some kind of internal application to be used within your office or similar. For this particular purpose, hosting the application via Webrick (the built-in web server in Rails) might be a sufficient solution. To do this, start the server with a -p
command line argument: ruby script/server -p 80
This obviously requires port 80 to be available (not bound by some other web server). Also, on most operating systems, you will need root privileges to bind to port 80. The security implications of running a web site as root are serious, so you really only want to do this if you know what you are doing, and are absolutely sure that the server is completely shielded from the Internet.