Wet coughs are those that have Phlem or mucous.  The object is not to supress this but to enable it to come out.  

There are certain herbs that have Expectorant properties to them.  You can make teas, syrups or cough drops with them.
 

Dry coughs are those nagging, unproductive coughs that won't go away.  Antitussive herbs are what works best for this.

I have added a few files to this page that you can download.  They are some cough remedies and a list of Expectorant herbs you can use for wet coughs.  I retrieved this information from:  http://www.anniesremedy.com/  This is a wonderful resource to information on herbs!