WP Ajax Login is a simple solution to add an ajax login form to your WordPress website.

Download the source files and drop the wp-ajax-login directory inside the root of your theme directory.

Include wp-ajax-login.php

In functions.php, go ahead and add the follow require_once to include the init script.

require_once ( get_template_directory() . '/wp-ajax-login/wp-ajax-login.php' ); // WP Ajax Login

The Shortcode

Place the following shortcode anywhere on your site to add to login widget.
[ wp_ajax_login loadingmessage="Logging in..." ]
Full documentation is avaialble on Github.