Display taxonomy term description when mouse hovers I created a taxonomy and added vocabulary with terms(each term has a description). Included the vocabulary to an auto-complete field. It all works good.

In the auto-complete field, terms appear in a drop-down menu as a list without any description. 

Here, I want to add a feature for the field in such a way that when the user hovers through the terms in the drop-down menu it should display its corresponding description.

And, Also have an idea of implementing the stackexchange Tags drop-down feature. I like the SE approach of displaying the tags (as buttons) and its corresponding description (below the button). I'm loving it. 

Either way is ok, displaying the description when mouse hovers and SE drop-down feature.
You could try using beautytips and views. There is an old Drupal 6 video explaining how to do it for feeds, showing a part of the body when hovering the title in a list: http://www.veoh.com/watch/v200239557fzYsENz.
It is still pretty much accurate on the conception of it.