Displaying just one post from each category on archive page

Don’t use query_posts() use the WP_Query class instead… the link to the Codex will also reference all the appropriate query parameters.

The following would query the most recent post in a particular category, you will need to define the $category_id

// The Query
$query = new WP_Query( 'cat' => $category_id, 'posts_per_page' => 1 );

// The Loop
if ( $query->have_posts() ) {
    while ( $query->have_posts() ) {
        $query->the_post();
        get_template_part( 'content-category', get_post_format() );
    }
} 

/* Restore original Post Data */
wp_reset_postdata();