Why would you want a short URL?
Well they are short and this in the world of Twitter is important because you only have 140 characters to play with. So if you want to put a web address into your tweet then you need to keep it as short as possible.
Most blogging tools will generate a short URL for you, in WordPress for example there is a “Get Shortlink” button at the top of your edit page (this article is http://wp.me/pEQlo-2A for example).
Google now offer a short link tool (you will need a google account) go to http://goo.gl/. This is quite a useful tool because you can create a short URL to anything on the web and keep a record of what you have created (see http://goo.gl/DA0A ).
If you are wanting to save space then Google wins by 3 characters (for the time being).