Ensure jquery is activated

In your function that hooks “wp_enqueue_scripts”, add this:

//********** ENSURE jQuery IS ALWAYS LOADED ON PAGES **********
//(we use in our custom code)
add_action( 'wp_enqueue_scripts', 'MyThemeEnqueueScripts' );
function MyThemeEnqueueScripts()
{
  wp_enqueue_script( 'jquery' );
}
add_action( 'admin_enqueue_scripts', 'MyAdminEnqueueScripts' );
function MyAdminEnqueueScripts( $hook )
{
  wp_enqueue_script( 'jquery' );
}

Simple Working Example

In functions.php

add_action ( 'wp_ajax_my_ajax_action_callback', 'my_ajax_action_callback' );        //For AJAX calls made for logged in users
add_action ( 'wp_ajax_nopriv_my_ajax_action_callback', 'my_ajax_action_callback' ); //For AJAX calls made from non logged in users (include both if both logged in and not logged in users will trigger this)
function my_ajax_action_callback()
{
  //Check the nonce
  check_ajax_referer('my-nonce-special-string', 'security');      //This will die(); if the nonce check fails

  //if (!is_user_logged_in())
  //  die();  

  //$user_id = get_current_user_id();                             //Use this if your function wants the wordpress user_id (secure method, don't pass it as an ajax argument)

    $ajax_value1 = 1234;
    $ajax_value2 = sanitize_text_field($_POST['my_value_1']);
    $ajax_value3 = 'abcd';
  
    $ReturnArray = array(
                    "ajax_value1" => $ajax_value1,
                    "ajax_value2" => $ajax_value2,
                    "ajax_value3" => $ajax_value3
                    );
    echo json_encode($ReturnArray);
  
    wp_die();     //Terminate immediately and return a proper response
}

In your php (html) page

  $AjaxUrlHtml = "var ajaxurl = '" . admin_url('admin-ajax.php') . "';";
  $AjaxNonce = wp_create_nonce( 'my-nonce-special-string' );
  
  $HtmlOutput .= <<<_END
  <script type="text/javascript" >
  $AjaxUrlHtml
  
  var Counter = 0;
  setInterval(function()
  {
      var post_data = {
                 'action': 'my_ajax_action_callback',   //The name of the ajax callback action in functions.php
                 'security': '$AjaxNonce',
                 'my_value_1': 9876
      };
      
      jQuery.post(ajaxurl, post_data, function(response) {
        //----- CALLBACK SUCCESS - RESPONSE RECEVIED -----
        response = jQuery.parseJSON(response);     //Decode the json response
        
        Counter++;
        document.getElementById('ajax_value1').innerHTML = Counter + '-' + response.ajax_value1 + '-' + response.ajax_value2 + '-' + response.ajax_value3;
      });
      
      
  }, 1000);		 //<<<<Call every time in mS
</script>

<p>AJAX Output: <span id='ajax_value1'>Retreiving...</span></p>

_END;