Snow crystals form by a process of sublimation. That's where water vapor turns to ice without first turning into a liquid form. Usually, a speck of dust or other tiny floating particle is needed to start the process. Something as small as a bacteria can even act as the trigger for snowflake formation. As the dust floats through the atmosphere, ice crystals form on the particle and begin to build into a snowflake.
Individual crystals connect to other crystals and ultimately form larger structures.
That's how snowflakes get started. It is said that no 2 snowflakes are exactly alike and this may be true. A photographer named Wilson Bentley took over 6,000 snowflake pictures and none of them were the same. Each one is formed at a different atmospheric temperature. The difference between one form and another might be as little as 1 degree of temperature.