There are several kinds of snowstorms and each can be substantially different that other types in many ways. However, all of them require two things: Humidity and cold temperatures near the ground. Let's look at how, in general, a typical snowstorm develops, ignoring for the moment the different kinds of snowstorm.
Warmer air, such as that laden with moisture, rises. As it does so, it releases heat and the air grows progressively cooler. This causes the moisture to begin to condense into larger and larger droplets, each forming around a particle, such as of dust. Partly because it is still less dense than the air above it, and partly because of the still warmer air below that is still rising, the column of air continues to rise.
It continues to get colder as it gains altitude. Clouds form and become thicker. This is the anatomy of virtually all clouds.
As more and more droplets merge, they become heavier. When they become so heavy that the air rising beneath them cannot support the weight, they begin to fall. At this point, the droplets may be many thousands of feet in altitude, where temperatures can be well below the freezing point of water.
As the droplets fall, they begin to freeze because of the temperatures. This is true even in the summertime. This is the reason temperatures near the surface of the earth are important. In the summer, in most places, as the frozen droplets get nearer to the ground and encounter warmer air, they melt and fall as rain.
When temperatures aren't warm near the surface, though, the droplets keep falling in frozen form and humidity that adheres to them forms a lattice like frost on a window, six sided and intricate. This is the snow most of us are used to. It doesn't have the chance to melt, so it continues to fall as snow. If it mostly melts then refreezes, this results in sleet or pellet snow.
It is important to understand, too, that if there is a thin but warm layer right at ground level, the snow may not have time to melt, which is the reason 'wet snow' is common in the mountains, especially when temperatures are between 35 and 40 degrees Fahrenheit. In fact, the largest snowflakes tend to fall at this temperature because as they begin to melt, they collide with other flakes on the way down, and combine with them. Enormous flakes may result.
This also explains and exposes some myths, such as that it can't snow if it is too cold. It can snow at even subzero temperatures, but the flakes usually don't fully form and there are few collisions, so the snow usually amounts to tiny frozen balls often referred to as 'sawdust snow'. This kind of snow is a snow skier's dream, when it is right on top of wet snow. However, owing primarily to its small size, it usually doesn't accumulate fast like wet snow often does.
There are many different kinds of snow. The formation of the snowstorm follows the same basic steps, though the variations can result in different snow formations. Having an idea of how a snowstorm develops though can let you anticipate the sort of snow to expect.
Watching the snowstorm develop can enhance the enjoyment of the snow, or to avoid the troubles. It is sure worth taking the time to get an understanding for the phenomenon.