Q: How are snowflakes formed?
A: A snowflake begins to form when an extremely cold water droplet freezes onto a pollen or dust particle in the sky. This creates an ice crystal. As the ice crystal falls to the ground, water vapor freezes onto the primary crystal, building new crystals – the six arms of the snowflake.
That’s the short answer.
The more detailed explanation is this:
The ice crystals that make up snowflakes are symmetrical (or patterned) because they reflect the internal order of the crystal’s water molecules as they arrange themselves in predetermined spaces (known as “crystallization”) to form a six-sided snowflake.
Ultimately, it is the temperature at which a crystal forms — and to a lesser extent the humidity of the air — that determines the basic shape of the ice crystal. Thus, we see long needle-like crystals at 23 degrees F and very flat plate-like crystals at 5 degrees F.
The intricate shape of a single arm of the snowflake is determined by the atmospheric conditions experienced by entire ice crystal as it falls. A crystal might begin to grow arms in one manner, and then minutes or even seconds later, slight changes in the surrounding temperature or humidity causes the crystal to grow in another way. Although the six-sided shape is always maintained, the ice crystal (and its six arms) may branch off in new directions. Because each arm experiences the same atmospheric conditions, the arms look identical.
Q: So, why are no two snowflakes exactly alike?
A: Well, that’s because individual snowflakes all follow slightly different paths from the sky to the ground —and thus encounter slightly different atmospheric conditions along the way. Therefore, they all tend to look unique, resembling everything from prisms and needles to the familiar lacy pattern. 
(information & pictures courtesy of NOAA)
