There are a lot of different reasons why you might want to have a page or URL removed from Google's search index. Some of the valid reasons include:
If a page contains confidential or sensitive information - Obviously, you never want pages such as these indexed by Google, let alone be accessible to the public.
If you have taken down a page permanently - Of course you can always do a redirect, but you can also just have the URL of that page removed from Google's index to ensure that it doesn't even appear in Google's SERPs.
If a page has contents that transgresses the law - Whether this page is yours or not, you can submit the URL of the page for removal. Of course, if the page is yours, it makes more sense to just take the page down!
If you need to have a URL removed for the above reasons, the best way to go about it is by using Google Webmaster's UL Removal Tool. The tool allows you to submit a URL to Google directly for removal so that you will get results much faster than if you simply tell Google not to crawl the site by indicating it in your robots.txt file.