jquery ajax function not working

Try this

    $("#postcontent").submit(function() {
  return false;
};

$('#postsubmit').click(function(){

// your ajax request here
});

put the event handler function inside $(document).ready(function(){...}). it shall work now

also add preventDefault() to restrict page refreshing

$(document).ready(function() {

            $("#postcontent").submit(function(e) {
                e.preventDefault();
                $.ajax({
                    type : "POST",
                    url : "add_new_post.php",
                    data : $("#postcontent").serialize(),
                    beforeSend : function() {
                          $(".post_submitting").show().html("<center><img src='images/loading.gif'/></center>");
                    },
                    success : function(response) {
                        alert(response);
                        $("#return_update_msg").html(response);
                        $(".post_submitting").fadeOut(1000);
                    }
                });
                e.preventDefault();
            });

        });

try this code

<script src="//ajax.googleapis.com/ajax/libs/jquery/1.11.0/jquery.min.js"></script>
<Script>
$(document).ready(function(){

$("#postcontent").click(function(e) {

        $.ajax({type:"POST",url:"add_new_post.php",data:$("#postcontent").serialize(),beforeSend:function(){
            $(".post_submitting").show().html("<center><img src='images/loading.gif'/></center>");
        },success:function(response){   
                    //alert(response);
                    $("#return_update_msg").html(response); 
                    $(".post_submitting").fadeOut(1000);                
            }
        });
   });
});
   </script>

<form name="postcontent" id="postcontent">
              <input name="postsubmit" type="button" id="postsubmit" value="POST"/>
              <textarea id="postdata" name="postdata" placeholder="What's Up ?"></textarea>
 </form>

you need to prevent the default behavior of your form when submitting

by adding this:

$("#postcontent").on('submit' , function(e) {

  e.preventDefault();

  //then the rest of your code
}