URL shortening is the process in which you take a long URL, such as this:
http://blog.penelopetrunk.com/2013/10/07/the-pursuit-of-happiness-makes-life-shallow/ and you make it into something simpler, such as this:
http://goo.gl/rHwPN0 (this is the same link as the above link, when you click it and you will be redirected to the long URL).
URL shortening started (I think) in the first years of 2000, and the most notable example of the first years was tinyurl.com. The web site still has a similar interface of what it used to have back then. Time passed, and more services appeared, such as bit.ly or goo.gl (and many many others, of course).
The current blog post is about URL shorteners – which one to choose, why should you use them, why avoid using them, some real life examples and some conclusions.