My understanding is that you have links on your website, and you want to be able to track the outgoing links on your site (links that connect to a site other than yours). I'm sure there are scripts/software you can find to handle this, and the idea would be along the lines of this:
Rather than linking "Twitter.com/Username" you could have "YourSite.com/Links.php?id=1". The script you write (or find) would know that id=1 means it should send the viewer to "Twitter.com/Username", but before it does this, it will add +1 to the hits, and log IP/browser/hostname/etc. Then you could create a backend page that lets you view the results.
I'm sure there are scripts that do this for you (and I believe Google Analytics has a way to implement this functionality). A quick Google search would definitely yield you some results on the subject. Hope this helps.