Remove URL from the Google Index

There has been repeated discussion about how and why you should remove a URL (whether it a whole website, directory or just one file) from the Google index.
The most common issue I have ran into is when two or more domains resolve to the same DNS resulting in more than one website with the exact same pages and content in the google index. I discuss this in one of my earlier posts "Problems with Multiple Domains".

Matt Cutts has presented us with a video explaining more scenarios and the best practices of how to remove URLs from the Google index even more in depth.