You can create a taxonomy-people_categories.php or a more general taxonomy.php (which will display all your taxonomies in this category – excluding post categories) – please see Taxonomy Templates / Template Hierarchy.
As far as a what you need to display your posts, once you have the taxonomy-people_categories.php created you can just run the normal WordPress loop:
<?php if(have_posts()) : ?>
<?php while(have_posts()) : ?>
<?php the_title(); ?>
<?php endwhile; ?>
<?php endif; ?>