Uncaught TypeError: $(...).dialog is not a function: How do I include dialog?
Drupal does not assign "$" to jQuery. That is because there may be libraries, which want to do something different with "$".

So you have to do this yourself:

(function ($) {
$( "#sponsor_dialog" ).dialog({
      autoOpen: false,
      height: 'auto',
      width:  520, resizable: false,
      show: {
        effect: "blind",
        duration: 1000
      },
      hide: {
        effect: "explode",
        duration: 1000
      }
    });

    $( "#sponsor_opener" ).click(function() {
      $( "#sponsor_dialog" ).dialog( "open" );
    });
})(jQuery);