My second-grader has been asked to write a "fairy tale." We are both clueless about what makes a story a "fairy tale" different from a fantasy story. Is there some special, defining element to a fairy tale that we're missing?
I believe that the traditional sense of the term fairy tale is used for a fairly concise story that is written to appeal mainly to children. The general context of a fairy tale would be the standard "Once upon a time.... and they lived happily ever after, THE END"
Generally these stories involved magic, fantasy characters and creatures, and were meant to teach a lesson, but not always.
Merriam Webster's dictionary defines fairy tale like this 1 a : a story (as for children) involving fantastic forces and beings (as fairies, wizards, and goblins) —called also fairy story b : a story in which improbable events lead to a happy ending http://www.merriam-webster.com/dictionary/fairy%20tale
Fantasy can include all of these, but is generally perceived to be a longer work of fiction and may or may not be geared toward a younger audience. You can have works of Fantasy that follow parallels of a fairy tale type story, but do not fit the traditional definition of a "fairy tale". Hope that helps.
There are several distinctive elements that differentiate fairy tales from modern fantasy, if we understand "modern fantasy" to mean "the modern literary genre established by authors such as J. R. R. Tolkien and C. S. Lewis, and the various sub-genres that have emerged from it since."
Scope: A fairy tale takes place close to home for the character. Jack plants a beanstalk in his own back yard. Red Riding Hood wants to go visit her grandmother, who lives within walking distance for a young girl. Sleeping Beauty, Snow White and Cinderella all got mixed up in the affairs of local rulers, and so on. But in modern fantasy, the journey is a central element. Bilbo travels from the peaceful Shire to a far-off mountain kingdom, and the generation after him goes much further, all the way to Gondor and Mordor. The children from The Chronicles of Narnia are transported from Britain all the way to another world, and they spend a great deal of time adventuring throughout one exotic locale after another within that world. Rand al'Thor's journeys take him to practically every notable location on an entire continent, and so on.
Location: A fairy tale takes place close to home for the (original) audience. Even if it's not explicitly set in the real world (but with magic), its setting tends to be a very familiar place. Modern fantasy, on the other hand, is very bound up with the art of worldbuilding. It almost always takes place on a different world, one that is alien in many ways. (Even with locations like Terry Brooks's Shanara, which is theoretically set on a post-apocalyptic Earth where magic somehow exists, it's sufficiently non-Earthlike that that's more of a framing device than anything.) Fantasy authors often tend to revel in seeing just how different they can make a world. For example, Brandon Sanderson's Roshar, the setting of The Stormlight Archive, is a world that's been ravaged by hurricanes, powered by magic so they don't blow themselves out over land, every few weeks for millennia. As a result, there's no such thing as soil; everyone lives on bare bedrock, with plants and animals adapted to such an alien ecosystem. You would never see a world like that in a fairytale!
Characterization: In a fairy tale, the protagonist is generally an audience surrogate, and the characters that he or she interacts with are archetypes. Just look at how many stories Prince Charming shows up in: he doesn't even need a name, because his archetype is who he is, and that rarely changes for any of the characters. Whereas in modern fantasy, character development and character growth is frequently a central theme. Compare Luke Skywalker the naive young farmboy to the fledgling Force-user who turns off his targeting computer to rely on the Force, the brash trainee who abandons Yoda and Dagobah to confront Darth Vader and rescue his friends, and the mature and confident Jedi Knight who can stand without flinching as Vader stands right behind him holding an active lightsaber. (Yes, Star Wars is very much a modern fantasy tale; the sword-and-sorcery elements are far more integral to the plot than the spaceships and computers.)
Magic: In a fairytale, magic is something that happens to the protagonist. Whether it be Snow White's poisoned apple or Cinderella's gifts from the Fairy Godmother, the magic is always external in nature and something the protagonist has to deal with. In modern fantasy, it's very common to have the protagonist be a magic-user themselves. Even in Tolkien's works, both Bilbo and Frodo actively wielded the power of the Ring to help advance their goals.
While not every story in one genre or the other will conform to all of these points, (Urban Fantasy in particular blurs the line quite a bit,) they make good general guidelines to distinguish between the two story types.
A fairytale is a more palatable way of demonstrating a moral or ethical lesson to your audience. No-one wants to be preached to but if the lesson is couched in an entertaining form - in this case a story - then they will listen and, despite themselves, learn.
The reason for the fantasy elements is part of this. By using otherworldly characters, creatures and settings the storyteller isn't pointing fingers or apportioning blame to any particular individual or group that the audience might know. This keeps things clear in the audience's mind and makes it much easier to use allegory to tell the tale.
Hope that helps.
These are probably evolving terms rather than hard and fast divisions, but what I would say distinguishes fairy tales from fantasy is that a fairy tale is a tale about a human being encountering fairy folk, who represent a danger to ordinary human life. Fairies have been Disnified in recent years, but Yeats poem "The Stolen Child" (http://www.online-literature.com/yeats/816/) is much closer to the original idea.
Fantasy, on the other hand, has become more and more about the wielding of magic or otherwise fantastical powers. Thus we have Harry Potter: not a story about ordinary people encountering dangerous magic, but about an extraordinary boy practicing magic and using it for good. In this sense fantasy is very close to science fiction, which is again about the exploitation of knowledge for power.
In this sense we may see the fairy story and fantasy as diametrically opposed takes on the same theme. And in this sense, horror today is perhaps closer to fairy stories than fantasy is.
In this specific case, I'd be very surprised if your child's teacher expected you to define it narrowly. Any fantasy story would probably be acceptable.
In general, however, a fairy tale is a short story with archetypal characters and plotlines. According to wikipedia the characters and settings are usually drawn from European folklore, however, there are similar kinds of stories from all around the world. The stories nearly always include some element of magic (spells, wishes), fantasy (little people, fairies, talking animals) or the supernatural (fate, prophetic dreams, genies), but there are stories which arguably fall in the category that don't (usually stories about tricksters or heroes of extraordinary cleverness). Fairy tales often seem to have deeper psychological meaning.
A typical fairy tale plot has a hero/heroine who is the youngest child, and who is abused, neglected, or considered worthless (or who is under a spell or curse). The hero will get supernatural aid (sometimes because of personal merit, sometimes not) to overcome a series of challenges, and will finish the story by becoming king or queen.