Widgets not showing in my custom theme

You’re using the widget name and not i.d which is why it isn’t working.

Change this:

if (is_active_sidebar( 'My_Widgtet_Area' ) ) : ?>

        <div id="widget-sidebar">
            <ul>
                <?php dynamic_sidebar( 'My_Widgtet_Area' ); ?>
            </ul>

To this:

if (is_active_sidebar( 'partner-slide' ) ) : ?>

        <div id="widget-sidebar">
            <ul>
                <?php dynamic_sidebar( 'partner-slide' ); ?>
            </ul>

The i.d in the template tag immediately above should match the i.d you use to register the widget in your functions file which in this case is partner-slide

register_sidebar(array(
    'name' => 'My Widget Area',
    'id' => 'partner-slide',

That’s one obvious problem and there may be others however you should follow the Codex as it provides working examples with related links http://codex.wordpress.org/Function_Reference/dynamic_sidebar