The compiler gives the error
Parse error on line 3: Unexpected 'IDENTIFIER'
referring to the line
dataType: 'html' error: (jqXHR, textStatus, errorThrown) ->
The problem is simply that there's no comma (or line break) between 'html' and error. Here's the fixed code:
$.ajax '/',
type: 'GET'
dataType: 'html'
error: (jqXHR, textStatus, errorThrown) ->
$('body').append "AJAX Error: #{textStatus}"
success: (data, textStatus, jqXHR) ->
$('body').append "Successful AJAX call: #{data}"
I highly recommend using an editor with a built-in "Build" command, especially one that can work on selected text. It makes syntax errors a lot easier to pin down.
coffee-script.js, you should see compile-time errors on your browser's console. If your browser doesn't have a console, grab the Firebug Lite bookmarklet.