Summarize:

When your nose runs, it may be due to sickness (which will likely be accompanied by other symptoms of a "cold," such as sore throat, cough, etc), being upset (when we cry, excess water from our tears drains through the nose), or cold weather (as our nasal passages are designed to heat the air before it reaches our lungs, and in order to do so your nose produces extra fluid in the cold weather). It may also be related to allergies, irritants in your environment (such as smoke), or a side effect of certain medications. When you breathe through your nose, your sinuses warm and moisten the air by swirling it around the mucous membranes that line the passages. This prevents you from irritating your lungs and throat with air that is cooler than your body temperature. Water is a by-product of this process and the excess runs down the back of your throat and out your nose. Your sinuses perform this function year round, but because of the temperature difference in cooler weather (especially in wintertime) it is more pronounced in the cold. Therefore, they are nothing to get overly worried about. They are so common, in fact, that sometimes it is called "skier's nose" due to the fact that nearly 100% of competitive snow sport athletes complain about runny nose! A runny nose from the cold is NOT related to an illness (and not related to the "common cold.") Although many people believe there is a link between cold weather and "catching a cold," this is thought to be due to spending more time indoors where people's germs are more readily passed to one another (and is not thought to be related so much to the cold outside).
Be aware of possible diagnoses. Understand why your nose runs when it's cold. Know that runny noses from the cold are extremely common.