How to style dragged element

An alternative, using only CSS, is to style the :active pseudo-class of the element to the desired drag style. The dragged copy will be created based on this state.

However, the original element will be kept with this style, as the browser seems to keep it :active until drop. To avoid this we can assign the style in an animation that runs for a very short period of time. Enough for the browser to copy the style but not too short. 0.1s seems enough for Chrome, Safari and Firefox.

https://jsfiddle.net/mLsw5ajr/

$('#draggable').bind('dragstart', function(e) {

  // http://stackoverflow.com/questions/2419749/get-selected-elements-outer-html
  var stuff = $(this).clone().wrap('<div></div>').parent().html();

  e.originalEvent.dataTransfer.effectAllowed = 'copy';
  e.originalEvent.dataTransfer.setData('stuff', stuff);
});

$('#draggable').bind('drag', function(e) {
  // here I want to style the ghost element, not the source...    
});

$('#droppable').bind('dragover', function(e) {

  if (e.originalEvent.preventDefault)
    e.preventDefault();

  e.originalEvent.dataTransfer.dropEffect = 'copy';
  return false;
});


$('#droppable').bind('dragenter', function(e) {
  $(this).addClass('over');
});

$('#droppable').bind('dragleave', function(e) {
  $(this).removeClass('over');
});


$('#droppable').bind('drop', function(e) {

  if (e.originalEvent.stopPropagation)
    e.originalEvent.stopPropagation();

  var stuff = $(e.originalEvent.dataTransfer.getData('stuff'));
  stuff.appendTo(this);
  return false;
});

#draggable,
#droppable {
  background: #ccc;
  color: #fff;
  padding: 10px;
  margin: 10px 0 100px;
}

#draggable:active {
  animation: drag_style .1s;
}

#droppable.over {
  background: #000;
}

@keyframes drag_style {
  0% {
    background-color: #fc0;
    color: #000;
  }
  99% {
    background-color: #fc0;
    color: #000;
  }
  100% {
    background-color: #ccc;
  }
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div id="draggable" draggable="true">
  <p>Drag me to my target</p>
</div>

<div id="droppable">
  <p>Drop me</p>
</div>

A problem I found in Firefox is that the element is kept :active if there is no event triggering by another element (like a drop). To fix this we could trigger a click outside the element.

Not too sure about other browsers however the dataTransfer object contains a property called mozCursor. This allows you to change the cursor in the drag state, however this is a Mozilla property.

https://developer.mozilla.org/en-US/docs/Web/API/DataTransfer/mozCursor

An example of using this can be found at the following location, the setting is changed on dragstart (set to use default 'arrow' cursor), dragover (set to use auto drag cursor (arrow with copy)) and dragleave (reset to use default 'arrow' cursor):

http://jsfiddle.net/YkaCM/4/

Try the answers to:
Javascript: How can I set the cursor during a drag & drop operation on a website?

Updated your dragover with the following:

$('#droppable').bind('dragover', function (e) {
  $(this).addClass('over'); // new

  ...

http://jsfiddle.net/YkaCM/1/

You can't style this directly as it is a bitmap/copy of what the element looked like when the drag started:

http://jsfiddle.net/2EvGP/

EDIT:

You can actually cheat to achieve this by briefly changing the style of the element when the drag starts: http://jsfiddle.net/LULbV/

$('#draggable').bind('dragstart', function (e){

  [Set style here]

  setTimeout(function(){
    [Reset style here]
  }, 1);

  ...

});

This works flawlessly in Chrome 19, and shows the style change depending on how you drag in Firefox 13. You would need to reset the dragged element's style on drop.

(Note I have a pretty fast computer, so I'm not sure if this hack would still work on slow machines)