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.
$my_taxonomy = 'institute';
$terms = wp_get_post_terms( $post->ID, $my_taxonomy );
echo term_description($terms->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->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.