Why do snowflakes form into hexagonal structures?
Ice grows in many forms.  As mentioned in the other answer, all of the ice we are going to observe is Ice Ih, but there are many other forms.  See this phase diagram of water:


Image courtesy of Cmglee on wikipedia

The different ice regimes grow different crystalline shapes.  Ice Ih grows hexagonal crystals and in certain regimes you can find triangular and cubic ice crystals.  The hexagonal shape is a consequence of the bond angles within the water molecule as it forms into a solid crystal lattice.  

This phase diagram says we'll experience Ice Ih between 0 C and -100 C and throughout tropospheric pressures.  This ice crystal is hexagonal, but within this crystal form there are many ice habits of crystal growth.  


Image used from Weatherwise magazine, AMS

The axes of this plot are supersaturation with respect to ice ($e/e_{si} &gt; 1$) and temperature.  All of of these crystals are hexagonal but some are long skinny hexagonal prisms and some are very thin and wide hexagonal plates.  The snowflake is a dendrite and these crystals grow between -10 and -22 C in and supersaturation with respect to liquid water.

What happens is that the hexagonal crystal has 6 vertices connecting its 6 edges.  These vertices produce an increased gradient in vapor (indeed, the sharper the angle, the stronger the vapor gradient becomes).  At high supersaturations vapor is quickly deposited in the areas of the enhanced vapor gradient and the arms of the dentrite form.  The particular shape of the dendrite will depend strongly on the vapor gradient it is experiencing, which in turn is influenced strongly by its current shape and the environment it is growing in.

References:


Lamb, D. and Verlinde, H., 2011: Vapor-growth of individual ice crystals. Physics and Chemistry of Clouds. Cambridge University Press, Ch 8.3, 342-369.