Quest
Hazel, Fiver, Bigwig and the other rabbits get caught up in a lot of adventures. But we're classing this as a quest story (rather than as an adventure story) because all of these adventures are very goal-oriented. They leave Sandleford Warren to go find a better place to live; they go to Nuthanger Farm and Efrafa Warren to get does; etc. The bunnies aren't just having adventures for the sake of adventures, but because they need to do something to survive.
Young Adult
Putting this book in the Young Adult category is a hard decision, and we know Adams faced this problem when he was trying to publish the book. On one hand, the characters are bunnies and it's told in a pretty simple style that's really good for kids. (In fact, as we mention in our "In a Nutshell" section, Adams started this story for his kids on a car trip.) But on the other hand, this book is full of death and violence, which might not be suitable for kids.
Even though adults love this book, we're going to call it Young Adult because the issues it deals with—like trying to find a place in the world to call home—are issues that young adults especially have to deal with. Also, it has bunnies.
Fantasy
Maybe rabbits really do talk to each other and tell stories about a great hero rabbit named El-ahrairah, but until we know that for sure, we're going to classify this as a fantasy. Also, there's the little issue of Fiver's and Hyzenthlay's supernatural abilities.
Put it this way: When the publisher Rex Collings accepted Watership Down, he wrote to a friend, "I've just taken on a novel about rabbits, one of them with extra-sensory perception. Do you think I'm mad?" (source). Not mad, just fantastic.
Folklore
Watership Down isn't folklore, but the stories about El-ahrairah definitely are. The narrator compares El-ahrairah to other folklore figures like Brer Rabbit, John Henry, and Robin Hood (5.11). Plus many of the stories contain familiar folklore events—like when El-ahrairah tries to make a deal with death (which is a common folklore plot). Most of this book isn't folklore, but these stories are so folkloric it hurts.