You can do basic email campaign tracking via Google Analytics. The featureset isn't nearly as complete as a real email tracking system, but its nice to have it integrated into your website tracking. GA has a set of five variables that you can use in campaign tracking (of any type). For email, I like to use three:
You'll want to tag every link in your outgoing email with those three variables. To see your clicks, visit the 'All Traffic Sources' report, and find the source/medium that you want to see (e.g. newsletter_051608 / email). This will give you the number of visits that came to your site because of the emails (remember that a visit is one person doing as much as they want and then being idle for 30 minutes). Unfortunately, there is no way to see overall unique visitors… if someone clicked on a link, waited 30 minutes, and clicked on another link, that would be two visits. There is no way to know that is only one unique visitor.
To see which links people clicked on, use the 'Segment' drop down to segment that report by 'Landing Page'. Note that Google only records the first page in a visit as the landing page; they don't track each click in the email. So if a user clicked a link, then went back to the email, and then clicked another link, that would only count as one visit, and only the first link they clicked on is in the landing page report.
So Google Analytics isn't perfect for tracking email, but for a free, integrated system it isn't half bad. Of course, you can track all kinds of campaigns using the five variables.