Unless the asp code is adding quotes, you will need to do so. Otherwise you aren't passing a string into the .html() function:
$(function(){
$('#drop-down-id').on('change', function () {
$("#partial-6").html('@Html.Partial("_Edit",Model.Groups[0])');
});
});
This will put the same code into the #partial-6 element each time the #drop-down-id element is changed. If you want to actually change the information you are putting in the #partial-6 element, you will need to make an AJAX call to return the partial from the server.
$(function(){
$('#drop-down-id').on('change', function () {
$.get('path/to/server-side.asp', { id : $(this).val() }, function (serverResponse) {
$("#partial-6").html(serverResponse);
});
});
});
This will create a GET request for a server-side script that is sent with the GET variable, id, which is set to the value of the drop-down that triggered the code.
Note that .on() is new in jQuery 1.7 and in this case is the same as .bind().