Shorten the link inside the banner with bit.ly.Then when you go to bit.ly you will see how many clicks you got.The easy way.There are also "banner trackers" out there.Just make a google search!
Yes it is possible to track banner ads on other site with the help of Google Analytic. We need to create customized URL for your banner ad with some URL parameter with the help of Google URL Builder and place that URL in your ad banner. You can see these things in Google analytics in traffic source section.