$args=array(
  'orderby' => 'name',
  //'parent' => '5',
  'order' => 'ASC',
  'hide_empty' => '0'
  );
$categories=get_categories($args);
  foreach($categories as $category) {
    echo '
    <div class="' . $category->slug . '">
        <h3>' . $category->name . '</h3>
        <img src="' . get_bloginfo( 'template_url' ) . '/images/categories/' . $category->slug . '_feature.jpg" alt="' . $category->name .'" />
        <p>' . $category->description .'</p>
        <h4>Explore</h4>
        <ul>';

        global $post;
        $args = array( 'numberposts' => 3, 'category' => $category->term_id );
$myposts = get_posts( $args );
foreach( $myposts as $post ) :  setup_postdata($post);
    echo  '<li><a href="' . get_permalink($post->ID) . '">' . get_the_title($post->ID) . '</a></li>';
endforeach;
    echo  '
        <li><a href="' . get_category_link( $category->term_id ) . '">More...</a></li>
        </ul>
        <br />
        </div>
    ';
}