Google's URL Removal Tool

Vanessa Fox of the Google Webmaster Central team announced that Google Webmaster Central now supports the easy removal of content from their index.
In short, if you login to your Google Webmaster Central account, you can easily remove content from Google.com, if you have verified the site. The blog post also explains new ways of requesting the removal of content that you do not have access to.
Since then, people have been using it but there has been a lot of confusion on how it works and doesn't work.A Google Group thread has dozens of posts with questions on why pages they remove, still appear. Or why it may take so long to remove a page.
Due to that, Susan Moskwa, of Google Webmaster Central, promised clearer instructions on the Google URL removal tool and also provided more details on what you should do to remove content from Google's index.