A simple way to load related posts using WordPress categories.

related-posts.php

<?php

// Save this in layouts/related-posts.php

global $related_posts_id;
global $related_posts_cats;

$args = array(
    'posts_per_page' => 2,
);

if (isset($related_posts_id)) {
    $args['post__not_in'] = (array) $related_posts_id;
}

if (is_array($related_posts_cats)) {
    $args['category__not_in'] = (array) $related_posts_cats;
}

$posts = get_posts($args);

?>
    <!-- related post wrapper here -->
<?php 

// Loop through the related posts markup here
foreach ($posts as $post):
    setup_postdata($post);

?>
        <!-- post markup here -->
<?php

    wp_reset_postdata();
endforeach;

?>
    <!-- close the related post wrapper -->

single.php

<?php

// Put this inside the loop
$categories = wp_get_post_categories(get_the_ID(), array('number'=>1));
$related_posts_id   = get_the_ID();
$related_posts_cats = $categories;

get_template_part('layouts/related-posts');