Simply put, a URL shortener is an HTTP Redirect that allows a short web address to redirect to a longer one. For example:
Full URL of this page:
Shortened URL using Bitly.com:
See the difference?
This is a common practice in social media and useful for a variety of purposes. Here's a few of them:
Since Twitter is limited to just 140 characters, using a short URL takes up less space in your post, leaving more room for your message.
URL shorteners are also used to provide a "cleaner" web address when the actual web address is very long and complex. For example, if you're sending a long, complex web address in an email, the recipient's email program might "break" the URL between two lines. When the recipient clicks on it only part of the long address is sent to the browser's address bar. This results in a "page not found" error and can be very confusing. A shorter URL can help avoid the "broken" URL.
Tracking: Some services such as Bitly.com provide useful marketing statistics. For example, you could use different "shortened" URLs for Facebook and Twitter posts, both going to the same page of your website. Bitly.com even stores your Bitmarks for you and keeps track of your inbound traffic.
Warning: Spammers can use URL shorterners too, so the usual advice applies. Don't click on any link in an email unless you are confident that the source is legitimate.
Shortened URLs are permanent and never expire.