There are quite a few snowflake structures but they are always based on a hexagonal prism, the structure shown in the upper left of the figure below, that can be flat or columnar. Small crystals generally have this basic form but can grow branches from the corners as they grow bigger more complex forms.
The Fernlike Stellar Dendrites (lower left in figure above) make for the best powder skiing (low desnsity).
How they grow bigger:
Ok, this can get complicated but we’ll keep it simple for now. Essentially water vapor in a cloud has to diffuse/travel (diffuse = http://en.wikipedia.org/wiki/Diffusion) to get to the ice crystal to condense onto the crystal and start growing a branch. Once a bump begins to stick out from the original hexagonal prism shaped ice crystal it is effectively shortening the distance water vapor has to travel to get to that crystal. Therefore the water vapor will condense on that bump before condensing on a flat surface of the crystal. The original bumps used are the corners of the hexagonal prism. This is how branching occurs. This happens over and over on the ‘bump’ and that bump becomes a branch of the snowflake. In turn, bumps on the original branch begin growing and this is how even more complexity is added to each branch ending up in the intricate snowflakes we see on Earth. The branches are called ‘dendrites’ which just means ‘tree-like’. As an ice crystal floats in a cloud it morphs again and again thru differing conditions and tempuratures until it becomes heavy enough to fall. As it falls it again morphs as it encounters even more different conditions and temperatures and continues changing until it hits the Earth.
A growing designer snowflake
Why do snowflakes have six sided symmetry?
To really answer this question we need to look at why the original ice crystal is a hexagonal prism. This basic structure comes from the ice crystal lattice. The water molecules in an ice crystal form a six sided lattice. This six sided-ness is simply the natural way in which water molecules arrange themselves in an ice crystal. From that initial hexagonal lattice, the ice crystal grows and continues with that hexagonal orientation. Check out the figure below where the red balls are oxygen and the gray bars are hydrogen. Two hydrogens for every oxygen making H2O.
Why is snow white if snow is made up of clear crystals?
A pile of fresh snow is basically a pile of crystals. All those crystals are reflecting the natural white light that hits them. Within the pile of snow there are many many different surfaces reflecting in many many different angles. The light bounces all over within the snow and then scatters back out of the snow into the atmosphere. All the colors of the visible light spectrum are scattered about equal giving snow it's white appearance. (Essentially snow absorbes hardly any of the visible light spectrum and thus reflects the same white light (which is made of all the colors of the visible spectrum) right back out into the world giving off a white appearance)
This is pretty crazy. Basically….. it's really complicated and you go here to learn more at snowcrystals.com