There are two major groups of fairytales: Tales in one group are based off of folk tales that existed for a long time and were retold and retold over a long period of time with no one knowing the original author. These tales are also likely to exist in several different forms, since they come from an oral tradition where things might have been intentionally changed to fit local circumstances, or simply changed due to not remembering correctly. The other group consists of tales that were written by known authors and are original ideas by that author. If they contain elements of folk tales, the author drew inspiration from those, but it would be considered plagiarism if they were completely and faithfully retelling a folk tale as their own.
Examples of the fairytales from the first group are of course primarily known today because at some point they were written down, and the authors who did are often known today. The perhaps most well known are the Brothers Grimm, but there are several others as well. These tales are tales like Hansel and Gretel, Sleeping Beauty, Little Red Riding Hood, Jack and the Beanstalk, Snow White and many, many more.
On the other side, the perhaps most well known author of fairytales who wrote original stories was Hans Christian Anderson. He wrote, among other things, The Snow Queen, The Little Match Girl, The Emperor's New Clothes, The Nightingale, The Ugly Duckling and The Little Mermaid. These are again only a small selection and he is by no means the only author to write such tales either.
Whether one or the other type however, fairytales usually have a fair few traits in common that allow us to identify them as fairytales.
Some of the more common ones are of course the magical numbers three, seven or nine. Almost everything in the story comes in one of these amounts, with three being the most common; three brothers, three tasks, three days, etc. There is also a clearly defined line between good and evil; fairytales are not the place to look for subtle distinctions between good and evil, or complicated characters who contain both. This is usually the case with the arc of the story.
There is one primary problem and the story is only concerned with this one thing and does not veer off and get side-tracked unless the side-track will later be revealed to be connected to the main plotline. They very often describe a journey of some sort, where the protagonist of the story starts out by leaving their home, then experience the bulk of the tale's problems on this journey, and only returns home after these have been resolved. Through the journey the protagonist has gained knowledge and has grown as a person. This too is important as most fairytales appear to want to teach a lesson that is important to the culture it came from. In Little Red Riding Hood for instance; be wary of strangers and do not dawdle in the woods.
There are many other elements of course, and if you are interested, one of the great places to look is Morphology of the Folktale by Vladimir Propp. This book has a very comprehensive list of all the various elements a fairytale may contain, although of course not every tale will contain all of them.
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