bitly (previously bit.ly, actually bitly.com) is a URL shortener that will make long web addresses incredible short. Users just need to enter the original URL and this online website will generate a very short version of it.
After shortening, the old and new URL are displayed, you can easily compare them and copy the short one to use it for different purposes (share on Twitter, send by email etc.).
Is a good alternative for TinyURL and provides API access for developers. You can keep a history of all generated URLs if you create an account.