Most of the time (unless you're watching a Victorian costume drama), coughs aren't usually signs of anything terribly sinister. They're a natural part of the common cold, and in most cases, they will clear up on their own. But where do they come from, and what do we do until they decide to go away?
Coughs are either caused by viral infections or symptoms of viral infections like bronchitis and flu. Because many of these viruses affect the respiratory system, they cause dry coughs as the brain tries to get rid of the "foreign objects" in the throat - which is what it perceives the infection to be.
Sometimes, infections respond by creating lots of mucus. At that point, the cough reflex triggers to clear the mucus from the lungs (which usually proves more effective than trying to clear a sore throat with coughing). If the cough is clearly coming from the chest by isn't full of phlegm, it may be because the lungs and airways are swollen and irritated.
Unfortunately, most of these illnesses don't have fast cures, so all you can do is find a way to wait them out as comfortably as possible. Although antibiotics can help in some situations, more infections that cause coughs are viral infections and won't be affected by medicine meant for bacterial infections.
Cough suppressants can sometimes be helpful, especially if you have a dry, non-productive cough. (These are also the best candidates for cough drops.) If you have a mucus-y cough that's clearing phlegm from the lungs, it's better not to suppress it. Instead, a decongestant or an expectorant can be used to minimize the mucus or help it to come up more easily.
General cough syrups are not effective for everyone and are usually sugar-based. They can coat the throat to relieve irritation, but they don't always manage to keep up the relief for a very long time. More cough syrups, especially those that can be bought over the counter, contain so little of the active ingredient that they hardly do any good at all for some people.