There are several folds and ridges at the back of the nose, on the way to the throat and windpipe, which are covered with mucous membranes. These membranes secrete mucus, a thin but slightly sticky liquid. As the air we breathe passes over these folds and ridges, it is humidified by evaporating water from the membranes and mucus, and many of the dust particles present in the inhaled air stick to the mucus so they cannot continue to the lungs. Sometimes, when our noses and lungs are irritated by something in the air, our mucous membranes will produce more mucus to help pick up and clean out the irritants. Unfortunately, the excess mucus and water makes you very uncomfortable and doesn't do any better as far as self-cleaning goes.
Viruses can also attack the mucus membranes and cause extra watering and extra mucus production.
This, in fact, is how the common cold occurs. Other people who breathe those droplets may be infected also, since the droplets will be caught on their mucus membranes. Once you've been infected by a particular cold virus you'll be immune to it forever, but there are hundreds of cold viruses around for you to catch, and there'll almost always be a few around that you haven't had yet. Since, as with the vast majority of viruses, there are no medicines we can use to kill cold viruses, when we get a cold we have to wait it out, possibly with medicines to relieve some of the misery. Unfortunately, about all we can do about coughs, colds, and runny noses is to relieve the symptoms a bit.