if you use the stop function when you intialise .sortable() you can ajax send the new order to a script, which (when you write it!) would save the new order to the database.
jQuery(function(){
jQuery("#sortable").sortable({
stop: function (event, ui) {
var new_order = jQuery(this).sortable('serialize');
jQuery.post( "http://www.thissite.com/wp-admin/ajax.php", { action: my_custom_ajax_save, order: new_order }, function( data ) {
console.log('ajax sent and response received');
});
}
});
})
If you give more context we could help with the actual save. For example, if its a post you could save to the postmeta database nicely, what exactly are you saving?