One reason for the demand might be the requirement for the popular microblog Twitter. The blog entries or Tweets as the in-crowd calls them have a maximum character size of 140. Much like that for a text message on your mobile phone, which has a limit of 160. So saving on characters by shorting the url you are linking to is quite obvious and helps you squeeze in some more stuff.
Now here comes the problem with shortening url's. Normally when you see a url you can hoover over it and see where it points to in the browsers status area (try it out on some of my earlier links), this way you'd know where you would be going. This is one way to tell if an e-mail you just got from your bank is genuine and not some phishing e-mail. With shortened url's you don't see this. The link you get when you hoover over them is the same as the one on the website.
Now why would that be so bad? Well mostly you'll be safe but say you get sent to some site with code that is harmful to your computer. EEEK!!! right? Exactly my point. Although your browser might be smart enough to see this coming, it won't always be. Common sense is the best tool to protecting you from any danger on the web (and probably in real life). An application like a virusscanner, anti-spyware, etc can only take you so far.
So how to protect yourself? Well tinyurl has a option called Preview Feature, this will give you the ability to see the entire url before you continue. It's cookie based btw. Another option would be the Firefox extension LongURL Mobile Expander. This will give you the long url when hoovering over it. Although this only works on certain sites it would seem.
So keep your eyes on those rather short but not so trustworthy url's out there.