One common notion is that machine-made snow is artificial. This is not really the case -- it's actually the same stuff that falls out of the sky, it's just created by a machine rather than by weather conditions. The machine works very differently than a weather system, but it accomplishes exactly the same thing. To understand how machines make snow, it's a good idea to first look at how snow occurs naturally.
Snow comes from water vapor in the atmosphere. Clouds form when the water vapor (water in gas form) in the atmosphere cools to the point that it condenses -- that is, changes from a gas into a liquid or solid. The droplets in a cloud are so light that the air in the atmosphere keeps them aloft. If the droplets get too heavy, they fall in the form of precipitation. If it is cold enough, this water vapor doesn't condense as liquid water droplets, but instead as tiny ice crystals. In most parts of the world, rain generally starts out as snow but melts as it falls through the atmosphere (it is very cold at cloud level, even in the summertime).
Oddly enough, water doesn't automatically freeze at "freezing temperature" -- 32 degrees Fahrenheit / 0 degrees Celsius. You have to cool pure water to a much lower temperature (as low as -40 F / -40 C) for it to lose enough heat energy to change form. Usually, however, water in a cloud does freeze around 32 F / 0 C because of the work of nucleators, tiny bits of naturally-occurring material that help water molecules coalesce. The nucleators attract water molecules, which reduces their energy to the point that they form ice crystals. The nucleators in snow crystals are just dirt bits, bacteria and other material floating around in the atmosphere. Water condenses onto the nucleator, which becomes the nucleus -- the center -- of the snow crystal.
As the snow crystal moves around the cloud, more water particles condense onto it and freeze into crystals. The collection of individual crystals forms a snow flake. As the snow flake grows heavier, it falls toward the earth. If it is cold enough the whole way down, the flake will still be frozen when it reaches the surface.