How do I get posts from category using the slug?

Replace your category parameter with category_name

<?php
    global $post;
    $args = array( 'numberposts' => 10, 'category_name' => 'cat-slug' );
    $posts = get_posts( $args );
    foreach( $posts as $post ): setup_postdata($post); 
?>

<divs with the_title() the_excerpt() etc ></div>

<?php endforeach; ?>

For more info: http://codex.wordpress.org/Class_Reference/WP_Query#Parameters

suppose you have category name 'ice cakes' and category slug as 'ice-cakes', then our code to retrieve post under category 'ice cakes' is as follows:

<?php
              $args = array( 'posts_per_page' => 3,
               'category_name' => 'ice-cakes' );

              $icecakes = get_posts( $args );
              foreach ( $icecakes as $post ) : setup_postdata( $post ); ?>
                  <li>
                      <a href="<?php the_permalink(); ?>"><?php the_title(); ?></a>
                  </li>
              <?php endforeach; 
              wp_reset_postdata(); ?>