you should be returning an apt response from your server page which handles the ajax request. I will love to have one responsecode/status which says whats the status of the operation.
$.post("someurl.php",function(data){
  if(data=="success")
   {
      alert("Successfully Updated");
   }   
   else
   {
     alert("Some error in processing!");
   }
}
and from your php you return "success" if your operation is successful and return "error" if it is not successful.
EDIT : Here is how you do it with jquery ajax
jQuery post is a shortform of jQuery ajax method with post Type
$.ajax({
  url: "youruel.php?querystring1=12&id=34", 
  success: function(data){
    if(data=="success")
       {
          alert("Successfully Updated");
       }   
       else
       {
         alert("Some error in processing!");
       }
  }
});
You may add the error event also to handle those too.