[jQuery] Simple jQuery Ajax - Using PHP Variables

I have consulted numerous books, and have searched online in vain to
find an answer to this simple question.

Here is a case:

I have a simple PHP script that I can run either in the browser or in
the shell by typing something something similar to this:
scriptname.php?variable=value

Now let's assume that the variable is a simple string, and not
something complicated as an array. Let's also assume that the output
of the PHP script, is plain xHTML and the jQuery Ajax will insert it
into a div or span id="results".

Can someone provide an idiots guide to this, and also some code?

Thank you,
Chris