WordPress priv/nopriv demo

gistfile1.php

<?php
/*
 * Plugin Name: Ajax Test
 * Plugin URI: trepmal.com
 * Description:
 * Version:
 * Author: Kailey Lampert
 * Author URI: kaileylampert.com
 * License: GPLv2 or later
 * TextDomain: some-plugin
 * DomainPath:
 * Network: false
 */

$Ajax_Test = new Ajax_Test();

class Ajax_Test {

    function __construct() {
		add_action('wp_ajax_ajax_test', array( &$this, 'ajax_test_cb') );
		add_action('wp_ajax_nopriv_ajax_test', array( &$this, 'ajax_test_cb') );

		add_action('wp_enqueue_scripts', array( &$this, 'load_jquery') );
		add_action('admin_enqueue_scripts', array( &$this, 'load_jquery') );

		add_action('wp_footer', array( &$this, 'footer_script'), 99 );
		add_action('admin_footer', array( &$this, 'footer_script'), 99 );
	}

	function ajax_test_cb() {
		die( 'hello world' );
	}

	function load_jquery() {
		wp_enqueue_script( 'jquery' );
	}
	function footer_script() {
		?><script>
		jQuery(document).ready( function($) {
			$('body').prepend( '<button id="ajax_test_btn">Test!</button>' );
			var ajaxurl = '<?php echo admin_url('admin-ajax.php') ?>';
			$('#ajax_test_btn').click( function() {
				$.post( ajaxurl, {
					action: 'ajax_test'
				}, function( response ) {
					$('body').prepend( '<h1>'+response+'</h1>' );
				});
			});
		});
		</script><?php
	}

}