Snowflakes are either a single ice crystal or an aggregation of ice crystals which fall as snow. Each flake forms around a dust particle in supersaturated air masses by attracting supercooled water droplets in the clouds which freeze and adhere in crystal form.
The complex shapes form as the flake moves through differing temperature and humidity zones in the atmosphere and as a result individual snowflakes differ in detail from one another, but may be categorized in eight broad classifications and at least 80 individual variants.
When snowflakes land and accumulate, they undergo a change due to changes in temperature and form into a snowpack.
Once a droplet has frozen, it grows in the supersaturated air. Supersaturated air is where air is saturated with respect to ice when the temperature is below the freezing point. The droplet then grows by water molecules in the air vapor are deposited on the ice crystal surface.
The individual ice crystals are usually hexagonal and although the ice is clear, scattering of light by the crystal facets mean that the crystals often appear white in color due to diffuse reflection of the whole spectrum of light by the small ice particles.
The shape of the snowflake is determined by the temperature and humidity at which it is formed. The most common snow particles are irregular in shape, but near-perfect one are more common in pictures because they are more visually appealing. It is unlikely that any two snowflakes are alike due to the estimated 10 quintillion water molecules which make up a typical snowflake which grow at different rates and in different patterns depending on the changing temperature and humidity within the atmosphere that the snowflake falls through on its way to the ground.
Snowflakes are initially symmetrical and six arms, or dendrites, then grow independently, and each side of each arm grows independently. As a result, most snowflakes are not completely symmetric. Studies suggest less than 0.1 percent of snowflakes exhibit the ideal six-fold symmetric shape.
The shape of a snowflake is determined primarily by the temperature and humidity at which it is formed:
Freezing air down to 27 °F promotes thin, flat crystals.
In colder air down to 18 °F, the crystals form as needles, hollow columns, prisms or needles.
In air as cold as −8 °F, shapes become plate-like again, often with branched or dendritic features.
At temperatures below −8 °F, the crystals becomes plate-like or columnar, depending on the degree of saturation.
Forms below the saturation line trend more towards solid and compact while those formed in supersaturated air trend more towards lacy, delicate and ornate. Many more complex growth patterns are also sometimes observed.
An excellent online resource for complete information on snowflakes and ice crystals is Snow Crystals.
How to take photos of snowflakes
Guide to snowflakes from Cal Tech University is also a wealth of information and probably the most complete site on flakes.