How can I replace the search results displayed by WordPress?

You can use template_include filter hook to check if the current call is a search call and if so include your own template in which you can do what ever you want :

add_filter('template_include','my_custom_search_template');

function my_custom_search_template($template){
    global $wp_query;
    if (!$wp_query->is_search)
        return $template

    return dirname( __FILE__ ) . '/my_search_template.php';

}