Snowflakes are extremely beautiful objects and it is amazing that such patterns form randomly. However, that is just how they form. As you probably know, a cloud is just water vapor (steam, if you like), and is composed of very tiny water drops. Now, because the cloud is high in the sky, where the air is thin, these water droplets get cold, below the freezing point of water. It turns out that absolutely pure water won't freeze by itself, even if you get below 32 degrees F, or 0 degrees C (though it will freeze on its own if you get it extremely cold). Instead, the water drops need to stick to something else, like a particle of dust. This is called a "seed". Only then does the water droplet freeze. Once this has happened, other water droplets will stick to the newly made ice-crystal, and also freeze, making it larger. Depending on the conditions, different shaped crystals will grow. In general, we see that snowflakes have a six-sided symmetry. This is because of the shape of a water molecule (two hydrogen atoms and one oxygen atom stuck together), the angle between the two hydrogen atoms is about 120 degrees. This is also the angle between two sides of a regular hexagon (six-sided figure where all the sides are the same length). So in a sense, water molecules like to "fit together" in six-sided shapes. However, the rest of the snowflake's shape is determined by the "growing" conditions and the shape of the seed. So every snowflake is different, and beautiful.

- Answer provided by Ask Dr. Universe