A URL Shortener is a piece of software that takes a long URL or internet address and makes it smaller.  For example, the following is a link to a posting on my blog.  

https://dougpete.wordpress.com/2011/01/21/exploring-the-conversation/

That's a pretty long URL to read to someone and have them get it correct.  With services like Twitter where you're limited to 140 characters, you can use up your quota just typing the address.  If I took the same URL and ran it through a shortener like bit.ly, it becomes:

http://bit.ly/hih807

You've got to agree that's considerably shorter.  

There are many shortening services available for your use.  The legitimate ones that I've found are listed below.