There are several types of snowflakes, but they all have 6 sides or points when they are formed.
The question is, why? Well, it comes down to chemistry and their molecular structure. Snowflakes form from molecules of water and as we all learned in chemistry class, the structure of water is H20...2 molecules of hydrogen and 1 molecule of oxygen. When water freezes, the arrangements of the bonds (hydrogen bonds) in the molecule take a hexagonal shape, the same shape as most snowflakes.
Now the question arises, what makes snowflakes different from regular ice, from a refrigerator for example. The main thing is, the ice that is formed on the ground or in your fridge is not entirely pure. However, in a cloud, when snowflakes form, they need to start out as a supercooled water droplet. This is a water droplet is 100% pure with not even a speck of dust or dirt in it. These droplets are the most important detail while forming dendrites and precipitation in general. To explain, regular water (tap water, ground water, etc.) freezes at 32 degrees F, or 0°C. But, supercooled water will not freeze until it is -40°F or colder! This is critical to the growth of snowflakes. Dendrites typically are formed in the Dendritic Growth Zone (DGZ) which is around -10 to -20°C, or -4 to -14°F. If there are no supercooled water droplets, then snow growth will not take place. However, if there are regular water droplets, they will turn to ice and the cloud will be "glaciated" which means it turns into ice crystals.
So, how do the individual snowflakes form? Well, when there is a supercooled water droplet within the DGZ, all you need is a trigger object, or a "condensation nuclei". Basically, it is dust, dirt, or any tiny particles for the droplet to "catch" so it can freeze and crystallize. Once the droplet connects with a condensation nuclei, it is no longer a pure droplet and instantly freezes, so the creation of a snowflake has now begun. Depending on where it is in the cloud, the temperature, and how much lift there is in the atmosphere (along with a few other things), will determine the size and what type of snowflake it ultimately becomes.
Snowflakes can come in 35 different shapes as well as an infinite number of different sizes. A few of these shapes are: Stellar Dendrite, Needle, and Prism. These develop at different temperatures in the atmosphere and have their own distinctive atmospheric conditions which helps them evolve. Without getting into too much detail, the snowflake that everyone knows and loves is the "Stellar Dendrite". This is the classic 6 branched snowflake with many other appendages coming out of each branch.
This photo is from WeatherWorks Meteorologist Simon Wachholz. With over 50 snowflakes in this picture, you can see that none of them are the same and they also come in many different shapes and sizes. One thing to note about this image is these snowflakes are heavily rimed. This occurs when a snowflake passes through a supercooled cloud and accretes tiny droplets, making them look fuzzy.
So, if there are only 35 different shapes of snowflakes, does that mean it is a myth that all snowflakes are different? Not necessarily. Snowflakes can form in the same atmospheric conditions, then continue to grow and aggregate. While each individual snowflake has 6 sides, they can continue growing different branches from each side. Also, they can merge with each other and create "dendritic clumps". We can see these clumpy snowflakes when it's a "wetter" snow and temperatures are closer to freezing. Now, it is nearly impossible to say that no snowflake in history has ever been recreated. There is just no simple way to know unless a person is lucky enough to examine every single snowflake to ever fall to the ground. But, that of course, will never happen.
Snowflakes are unique in their own way and most of them are different from each other. The odds that you see 2 exactly identical snowflakes are nearly 0%. With the trillions upon trillions that fall during every snowstorm, good luck finding 2 of the same snowflakes, they will all have a different sequence to them, different shaped branches, or even different shapes all together.
*But* if you want to have some fun and experience the flakes for yourself, the next time it snows, freeze a black towel or cloth, go outside and catch them! Look at the beauty of mother nature, you never know what you will find.