<p>How do I preview a url using ajax? I have seen this done with search engine plug ins and would like to learn how to do this. Specifically, I would like to be able to mouse over a link and see the preview of the webpage using ajax.</p>

There's the
easy
solution, the
hard
solution, and the
use-a-library
solution.
use-a-library
: I prefer always doing the use-a-library solution unless you have a darn good reason otherwise.  One possible site which wraps the "hard solution" as a service for you:
http://thumbnails.iwebtool.com/demo/
easy
: The easy solution is to just load the target webpage as a downscaled AJAXy window.  You can use many of the Lightbox-class plugins for this task, particularly the ones which allow you to target arbitrary HTTP content for the Lightbox window.
GreyBox
is my favorite of those which I have used before.
Lightbox Gone Wild
is also nice.
hard
: Then there is the hard solution: you need to render the web page server side, cache the rendering as an image, and then serve up that image using Lightbox-esque Javascript (which is trivial next to the other requirements).  How you would go about doing this is outside the scope of this box.  Why would you do it this way?  The preview generates MUCH faster for the client, and it hermetically seals the client's session away from things which might bust it in the target website -- poorly behaving Javascript and/or malware can cause Really Bad Things when you open them, even in an AJAXy window-within-a-window.