Do you:
Have Java installed on that particular machine? If not, download and install Java from
www.java.com. Make sure to get Java SE v6.x. There are still a few kinks in v7. (This is usually obvious because you'll get a pop-up window saying something about an unknown file type.)
Have permissions to run Java applets? If you're running on a school computer, work computer, or public computer, you might not have those permissions.
Have any software running such as a firewall or anti-virus programs that would block you from running an app that doesn't come with whomever's certificate of security and approval? you usually get pop-ups from those programs, though.
Finally, is the file you're clicking on attempting to run named "autohaven.jnlp"? If not, it's not the right file. Try clearing out your Java cache and downloading again.
Even if those fail, you might be able to get one of the third party clients to run successfully. These are actually the preferred clients by the players and devs. You can find them in the "Wizards' Tower" forum. Ender's client is the most updated and popular. The forum:
viewforum.php?f=27
Opinions expressed in this statement are the authors alone and in no way reflect on the game development values of the actual developers.