Here's three URL shorteners that will allow you to get a shortened link without needing authentication, and with very simple APIs. These are useful if you're writing blogging or microblogging (twitter-style) applications.
All three take a simple HTTP GET request. As you have to include the URL you want shortened in the request, you have to HTTP encode it (you know, all that %20 stuff). All of them return one line of text, which is the shortened version of the URL.
You should be able to test them by simply using a webbrowser and going to the link. Don't forget to replace the line phrase <url goes here> with an http-encoded url.
Cli.gs http://cli.gs/api/v1/cligs/create?appid=autoblog-0.1&url=<url goes here>
Linkee http://api.linkee.com/1.0/shorten?format=text&input=<url goes here>
Is.gd http://is.gd/create.php?format=simple&url=<url goes here>