All definitions are my own based on my own experiences. If you don’t agree with anything, please say so in the comments. Once again, nothing said here is final and subject to change per your own experiences and thoughts. The examples are mostly taken from my bookshelf.
What is perhaps the most common type of fantasy, and what I think about when you say fantasy. Usually involves a journey or quest across a fantastical land. Also shares tropes with the space opera subgenre. Magic is typically a skill for one or two main characters.
The Lord of the Rings, the Inheritance Cycle, Brave Story, The Kingkiller Chronicles,
Much like how speculative fiction is the academic way of saying science fiction, Mystical Realism is the academic way of saying fantasy. The story is usually based more on the characters and their journey than on a plot. AS a result, there is less world building. I have the impression that stories considered mystical realism crossover with urban fantasy.
Fantasy written for a teenage audience. Characters are often teenagers who deal with teenage problems, often during a bigger issue within the book.
Cassandra Clare, Tamora Pierce, Kristin Cashore
Fantasy tropes and/or magical elements in a modern day or urban setting. Modern is a loose term, when you think about it. Urban fantasy that takes place in the fifties can potentially be considered a historical fantasy, so there is some crossover with that.
Eyes to See, The Dresden Files, Twilight, Sherrilyn Kenyon
Fantasy tropes and/or magical elements in a historical and/or period setting. Popular eras are the Victorian period or either World War. However, I do know of one book that takes place before the Revolutionary War, though.
Lady Lazarus, Dark Victory, Thieftaker
An exploration of the things that go bump in the night and/or an exploration of fears. Often features modern day legends such as vampires, werewolves, spectres. Often mixed in the urban fantasy genre. You might think this is its own genre, but I would file it under fantasy.
Blood and Other Cravings, Eyes to See,
Retold Fairy Tales
A retelling of a fairy tale of any kind. Always good fun.
Cinder, The Princess and the Hound, Tangled, Beauty