Most precipitation starts out as snow. The rain we observe on the ground is generally snow that has melted on the way down. Other types of precipitation, such as sleet and freezing rain, are caused by precipitation particles that melt and then encounter below-freezing temperatures again.
The formation of this precipitation relies on two phenomena. The first is the scarcity of tiny particles in the air on which water can freeze. The second is the fact that water vapor that is saturated with respect to liquid is supersaturated with respect to ice. This means that in a supercooled cloud, the few ice particles will grow rapidly at the expense of the liquid water droplets and begin falling as precipitation.
Two processes unfamiliar to us locals that produce snow (and rain) are upslope flow, where the air is forced to ascend by the topography, and lake-effect snow, where the air becomes unstable through the addition of heat and moisture from the warm Great Lakes.