Snowflakes grow from a basic prism to elaborate, lacy forms because the six corners continue to grow faster than the rest of structure. In a phenomenon called branching instability, water molecules diffuse through the air to reach the growing crystal, and reach the projecting corners first.
They build six branches, and projections on those branches soon sprout side-branches. The resulting crystal is called a stellar dendrite. The branching part of a nerve cell is also called a dendrite, which comes from the Greek word for "tree."
(Technically speaking, a snowflake is a conglomeration of crystals, which is how most snow falls to the ground. But the term is often used synonymously with "snow crystal.")