Show posts from one category with thumbnail and description on latest

I don’t know of any widgets or plugins that do this by default. To create this within your theme you would need to write multiple query loop that goes through each category then queries the latest posts from each with a counter so the top post can be styled differently.

Quick example.

$cats = get_categories();

foreach ( $cats as $cat ) {  //Loop through all the categories
    $count = 0;
    $args = array(
           'cat' => $cat->term_id,  //uses the current category in the loop
           'posts_per_page' => 4,
           'no_found_rows' => true,  //Performance optimizes the query
           'update_meta_cache' => false,  //We don't need post_meta
           );

    echo '<div class="aside-'. $cat->slug .'">'. $cat->name .'</div>';

    $cat_q = new WP_Query( $args );
    while( $cat_q->have_posts() ) : $cat_q->the_post();
    $count++

    if( $count == 1 ) { ?>  //Sets the output for the top post
        <div id="post-<?php the_ID(); ?>" <?php post_class('feature'); ?>>
            <fig><?php the_post_thumbnail(); ?></fig>
            <h3 class="post-title feature><a href="https://wordpress.stackexchange.com/questions/58240/<?php the_permalink(); ?>><?php the_title(); ?></a></h3>
            <p><?php the_excerpt(); ?></p>
        </div>
        <div class="right-side">
        <ul class="post-list">
<?php } else { ?>
        <li><a href="<?php the_permalink(); ?><?php the_title(); ?></a></li>
<?php }
     endwhile; ?>
         </ul>
         </div><!-- /end .right-side -->
         </div><!-- /end .aside-<?php echo $cat->slug; ?> -->

<?php } //Ends our category foreach loop