So you're enjoying spending some time on Twitter and catching up on the latest trending topic when someone you've never heard of before sends you a shortened link - Do you click it?
The security implication of shortened urls, such as tinyurl and bitly is that they hide the destination of where the link is going. They can also, as in the case of tinyurl's custom alias feature, potentially provide a misleading looking link.
The danger therefore is that it could be redirecting your browser to a malicious website that you wouldn't normally visit.
It's also important to realise that a link within a web page might not take you to the location that it says. This is common in phishing emails, whereby the displayed text appears to take you to a legitimate site, but in fact will go to a fake site with the aim to collect your personal details.
It's best therefore to try an use all the available information at your disposal to determine where you are likely to end up before you click on the link. For example, tinyurl provides a system on their website that allows you to check the destination of one of their links. Other methods include looking at the status bar when hovering over the link - try it with the itv site above - the actual destination is somewhere different!
As often in life, its best to look before you leap!