Remove Category Description Field

add_action(
    'admin_head-edit-tags.php',
    'wpse152619_edit_tags_trim_description'
);
function wpse152619_edit_tags_trim_description() 
{
    add_filter(
        'get_terms',
        'wpse152619_trim_description_callback',
        100,
        2
    );
}

function wpse152619_trim_description_callback( $terms, $taxonomies ) 
{
    if( 'product_cat' == $taxonomies[0] ) {
        foreach( $terms as $key => $term ) {
            $terms[$key]->description =
                wp_trim_words(
                    $term->description,
                    12,
                    ' [...]'
                );
        }
    }
    return $terms;
}

If you just want to truncate the description, in those cases when it’s long but you still want to see it in the admin table, use this:

/**
 * Remove default description column from category
 */
function jw_remove_taxonomy_description($columns)
{
    // only edit the columns on the current taxonomy, replace category with your custom taxonomy (don't forget to change in the filter as well)
    if (!isset($_GET['taxonomy']) || $_GET['taxonomy'] != 'category') {
        return $columns;
    }
    // unset the description columns
    if ($posts = $columns['description']) {
        unset($columns['description']);
    }

    return $columns;
}
add_filter('manage_edit-category_columns', 'jw_remove_taxonomy_description');