An unproductive cough is just a cough that doesn't produce any mucus or phlegm when you cough.
When you have a productive cough it means that the productive cough is producing mucus, phlegm etc during a cold or other virus and infection.
When you have a cold and have mucus and phlegm or another virus that has mucus or phlegm in your throat then coughing up that phlegm or mucus is good for you.
It means that you're getting rid of the bacteria and other viruses inside your body.
When you have the mucus or phlegm in your throat you should not swallow it because when you swallow phlegm or mucus it can lead to further infection.