The "Drop Down Menu" is a plugin. As the name suggested, it is supposed to be used to add a menu within an article. This is useful when you have a long article and wants to add a simple table of content or navigation menu to the long article (as what I've done in some of my articles).
If you think the "Drop Down Menu" works for your site, and you want to put it in a module position, you can use the method as outlined in the following article:
How to create a module with DirectPHP plugin (for Joomla 1.5.x)
The article is inteded for the DirectPHP plugin, but the same method will work for the "Drop Down Menu" plugin too (in fact, any plugin that you desire). The key thing is you need to install the module "CustomContent" by Ian MacLennan first.
Having said that, let me emphasize again (which I've also mentioned many times in some of the other replies), if you want a good menu system for your Joomla site, that menu should be part of your Joomla template.