To display shortcodes in widgets , use a regular text widget, and add this line to your theme’s functions.php file:
// Use shortcodes in text widgets.
add_filter('widget_text', 'do_shortcode');
This tells WordPress to render the shortcodes in widgets.
and if you want to call your shortcode by php you can use do_shortcode() function
// Use shortcode in a PHP file (outside the post editor).
do_shortcode('
');
To get a better understanding of shortcodes check out this tutorial which covers: What shortcodes are, How to use shortcodes, ,How to define your own. and How to use them in widgets.
Hope this helps