Okay... here goes...
Cough is a defensive mechanism which involves a cycle of deep
inspiration followed by an abrupt explosive expiration of
air.
Prior to understanding anything further, the respiratory tract are
full of defensive structures that helps prevents and filters
foreign bodies or particulates from the air that we breathe in. One
of such is the coughing mechanism, the other familiar principle, is
the sneezing reflex. Much like the gastrointestinal tract where
there is acid in the stomach and alkali in the intestines to
neutralize the necessary bacteria that attempts to breach our
immune system.
The difference between the cough and the sneeze reflex (aside from
having different nerve endings) is that cough is more for the
clearance of foreign bodies that have reached the lower respiratory
tract whereas sneeze is more for the clearance of the upper
respiratory tract (explaining why we can sneeze with our mouth
shut, but we cannot cough it out with our mouth shut).
There are two types of cough, the dry cough and the wet
cough.
Dry
cough - no phlegm is produced, one have the urge to
continue coughing as they can FEEL something in their throat - the
FEEL mechanism is that the tumor or the obstruction is encroaching
towards the nerve endings that is responsible for the coughing
mechanism.
Usually indicates heatiness, obstruction in the airways or
tumor.
Wet
cough - phlegm is produced, and there are many colors, drs
usually ask what color to indicate what kind of problem it is, if
it is green (bacterial infection), yellow (allergy, or infection),
red (blood) and white (mucus).
These usually indicates upper respiratory tract infection (URTI),
commonly the flu, and lower respiratory tract infection (LRTI),
commonly pneumonia, lung abscess, bronchiectasis or asthma (not an
infection, more of an allergy).
In summary, since coughing is a defensive mechanism for your body
in an attempt to get something out that is stuck in your throat,
most doctors would not advocate treatment by giving cough
suppressants, unless it is a dry cough (by means that there is no
point coughing, since you won't be able to cough up the tumor) OR
after abdominal surgery (the act of coughing, which reinforces the
abdominal musculature might open up the stitches)
And since the above explanation related to you, tells you of the
nerve endings that are responsible for initiating cough, these
cough suppressants will enter the brain through the blood brain
barrier and enter a certain structure called the cough center in
the brain. It will suppress the cough center and increases the
threshold for coughing. Since the sleep center is also nearby the
cough center, and given that most cough drugs will stimulate the
sleep center, this explains why we often feel drowsy after taking
these cough mixture and given the advice not to operate machineries
or drive after taking the medicine.
I hope the above explanation helps. I have long left med school, so
my explanation would not be that book based anymore, so sorry if
you find the text quite not by the book 
