<div class="checker-tag">
     
<h2>Form Title</h2>

</div>



<div class="col-md-6">
               
                
<form id="theForm" name="pc" method="post">
                        <input type="text" id="pc" name="postalcode" placeholder="Enter your postcode" value="">
                        <!----- the_ajax_hook is name of action ---->
                        <input name="action" type="hidden" value="the_ajax_hook" />
                        <button type="button" name="submit" class="btn btn-default hvr-sweep-to-right" onClick="submit_me();">Submit</button>
                    </form>

</div>

<!------ Response will display here -------->

<div id="response_area" class="col-md-6"></div>

Use admin-ajax.php file to call ajax

function js_enqueue_scripts() {
    wp_register_script("my-ajax-handle", plugin_dir_url(__FILE__) . "ajax.js", array('jquery'));
    wp_enqueue_script("my-ajax-handle");
    //the_ajax_script will use to print admin-ajax url in custom ajax.js
    wp_localize_script('my-ajax-handle', 'the_ajax_script', array('ajaxurl' => admin_url('admin-ajax.php')));
}

add_action("wp_enqueue_scripts", "js_enqueue_scripts");

Now, execute function

function the_action_function() {
 global $wpdb;
    $pc = $_POST['postalcode'];
    $sql = "SELECT * FROM `postcodes` INNER JOIN authorities WHERE `postcode` LIKE '$pc'";
    $results = $wpdb->get_results($sql);
    foreach ($results as $row) {
      //Execute Code Here
    }
}
// the_ajax_hook is now use here
add_action('wp_ajax_the_ajax_hook', 'the_action_function');
add_action('wp_ajax_nopriv_the_ajax_hook', 'the_action_function'); // need this to serve non logged in users

Now, Add js in ajax.js file

function submit_me() {
    jQuery("#loader").show(); //It will show loader div
    jQuery.post(the_ajax_script.ajaxurl,
                jQuery("#theForm").serialize(), 
            function (response_from_the_action_function) {
                jQuery("#loader").hide();                
                if (response_from_the_action_function == '') {
                    alert('Please check your postcode.It may be wrong or incorrect.');                    
                }
                else {
                    jQuery("#response_area").html(response_from_the_action_function);
                }
            }
    );
}

You can use also below format for js

jQuery('#theForm').on('submit', function (e) {
    jQuery('body').addClass('loading');
    e.preventDefault();
    var formdata = jQuery("#theForm").serialize() + "&action=" + 'the_ajax_hook';
    jQuery.ajax({
        url: the_ajax_script.ajax_url,
        type: 'post',
        data: formdata,
        success: function (response) {
            jQuery('body').removeClass('loading');
            if (response == 1) {
                jQuery("#response_area").html(response_from_the_action_function);
            }
        }
    });
    return false;
});
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