I assume you don't have full access to your server (e.g. shared hosting). In this case, have a look at deployhq.com or ftploy.com.
If you do have access to your server, you can use GitHub's post-receive hooks. See this StackOverflow question for more information.
Additionally, if you are deploying to Heroku/DigitalOcean/AWS and don't want to bother setting up everything manually, you can use dploy.io. (First repo is free)