Description of a sub-taxonomy

Term description works only with term id and the taxonomy name, so if you want to get the sub term’s description, you should first get all the children of a term(and their children, if its another level deep) and loop over their ids with a single term_description call per id.

<?php 
    $my_taxonomy = 'institute';
    $terms = wp_get_post_terms( $post->ID, $my_taxonomy );
    echo term_description($terms[0]->term_id, $my_taxonomy);

    //Assuming you have only 1 parent term, if multiple then loop over the $terms array
    $term_children = get_term_children( $terms[0]->term_id, $my_taxonomy );

    foreach($term_children as $term_child)
    {
        echo term_description($term_child->term_id, $my_taxonomy).'<br />';
    }
?>

This should give you an idea.