Fantasy is fiction which contains elements which can't, or don't, exist in the real world, often including elements of magic. In fact, some people say if there's no magic it's not fantasy.
There are numerous sub-genres under the Fantasy umbrella, some of the more well-known are:
High Fantasy – probably the most easily identifiable. Where the story takes place in an entirely fictional world, involving heroes / heroines, battles and quests. Example: Game of Thrones and Lord of the Rings
Low Fantasy – where the story is set more or less in the real world but involves magic or magical elements. Example: Harry Potter and the Philosopher's Stone
Paranormal Fantasy – where the story is set in the real world but supernatural creatures exist. Example: Twilight and True Blood
Dark Fantasy – a blend of fantasy and horror, where the horror element is through monsters or supernatural threats, instead of other humans. Examples: Coraline and American Gods
Like many subgenres, these often overlap and you can enter into some interesting debates on where your favourite story sits