This is really simple by using Drupal ajax framework.

First: Define your menu callback page:

/**
 * Implements hook_menu().
 */
function yourmodule_menu() {
  $items = array();
 
  $items['yourmodule/callback-page'] = array(
    'title' => 'AJAX callback - refresh the elements on the right hand',
    'page callback' => 'yourmodule_refresh',
    'page arguments' => array(1),
    'access arguments' => array('access content'),
    'type' => MENU_CALLBACK,
  );
 
  return $items;
}

The second: Do ajax deliver on callback page.

/**
 * Menu callback.
 */
function yourmodule_refresh($action, $type) {
  if ($type != 'ajax') {
    // This is js only.
    return 'Oh well';
  }
 
  $content = t('What ever you want');

  $commands = array();
  $commands[] = ajax_command_replace('.right-hand-wrapper-class', $content);
  $page = array('#type' => 'ajax', '#commands' => $commands);
  ajax_deliver($page);
}

This is really simple by using Drupal ajax framework.

First, define your menu callback page.

/**
 * Implements hook_menu().
 */
function yourmodule_menu() {
  $items = array();
 
  $items['yourmodule/callback-page'] = array(
    'title' => 'AJAX callback - refresh the elements on the right hand',
    'page callback' => 'yourmodule_refresh',
    'page arguments' => array(1),
    'access arguments' => array('access content'),
    'type' => MENU_CALLBACK,
  );
 
  return $items;
}

Secondly, do ajax deliver on callback page.

/**
 * Menu callback.
 */
function yourmodule_refresh($action, $type) {
  if ($type != 'ajax') {
    // This is js only.
    return 'Oh well';
  }
 
  $content = t('What ever you want');

  $commands = array();
  $commands[] = ajax_command_replace('.right-hand-wrapper-class', $content);
  $page = array('#type' => 'ajax', '#commands' => $commands);
  ajax_deliver($page);
}