I am using searchform.php and search.php files as already mentioned, but here I provide the actual code.
Creating a Search Page codex page helps here and #Creating_a_Search_Page_Template shows the search query.
In my case I pass the $search_query arguments to the WP_Query Class (which can determine if is search query!). I then run The Loop to display the post information I want to, which in my case is the the_permalink and the_title.
Search box form:
<form class="search" method="get" action="<?php echo home_url(); ?>" role="search">
<input type="search" class="search-field" placeholder="<?php echo esc_attr_x( 'Search …', 'placeholder' ) ?>" value="<?php echo get_search_query() ?>" name="s" title="<?php echo esc_attr_x( 'Search for:', 'label' ) ?>" />
<button type="submit" role="button" class="btn btn-default right"/><span class="glyphicon glyphicon-search white"></span></button>
</form>
search.php template file:
<?php
global $query_string;
$query_args = explode("&", $query_string);
$search_query = array();
foreach($query_args as $key => $string) {
$query_split = explode("=", $string);
$search_query[$query_split[0]] = urldecode($query_split[1]);
} // foreach
$the_query = new WP_Query($search_query);
if ( $the_query->have_posts() ) :
?>
<!-- the loop -->
<ul>
<?php while ( $the_query->have_posts() ) : $the_query->the_post(); ?>
<li>
<a href="<?php the_permalink(); ?>"><?php the_title(); ?></a>
</li>
<?php endwhile; ?>
</ul>
<!-- end of the loop -->
<?php wp_reset_postdata(); ?>
<?php else : ?>
<p><?php _e( 'Sorry, no posts matched your criteria.' ); ?></p>
<?php endif; ?>