I want me and any other developer to develop on our local machine, and push to GitHub. But I want to be able to have our production server update from the master branch of our repository on GitHub.

This is the solution I found:
http://mikeeverhart.net/git/using-git-to-deploy-code/

This solution asks I create a repo on the production server, and setup a post-recieve hook to perform a checkout to the web root.

However, with this solution the production server does not know about my GitHub at all.