Is Fantastic Beasts a prequel to Harry Potter?

The movie is roughly based on the book of the same name, which is a fictional textbook that's used by Hogwarts. The movie is essentially the story of the guy who wrote the textbook, and is not directly related to the main Harry Potter story.

Rowling has specifically said that the movie (which is supposed to be a trilogy) is neither a prequel nor a sequel to the main series.

Now that the movie has been released, we can complement KutuluMike's answer.

Tl;dr : the new Fantastic Beasts saga is more a

Dumbledore origins

than a prequel to the Voldemore saga.

As said in the accepted answer, the 2016 movie is independent from the Harry Potter saga. In particular, people that never watched or read Harry Potter can fully enjoy the movie.

However, it has now been announced by JKR that it will be a 5-parts story and that

a young Albus Dumbledore will feature heavily in the other movies.

Indeed, the main villain in FB

is revealed to be Gellert Grindelwald, the dark wizard that tried to create a wizarding empire and enslave Muggles, which lead to the global wizarding war. Albus Dumbledore has been very close to Grindelwald in his young age, developing feelings of love towards the dark wizard.

It is not a detail that he is portrayed by a A-lister. The main character of FB, Newt, does not seem connected at first to the main Harry Potter saga. But the family name of a supporting character of Harry Potter is dropped during the movie:

He had a relationship with Leta Lestrange.

It may just be an Easter Egg. More importantly, Newt has been expelled from Hogwarts before the events of the movie, for unknown (at least to me) reasons. He got support by only one professor,

Albus Dumbledore.

To answer this question properly, we have to have a really clear understanding of what exactly a prequel is.

In the question What makes a movie a prequel?, discussing whether Prometheus is a prequel to Alien or not, we get the following:

Prequel: A story or movie containing events that precede those of an existing work.

According to this rather vague definition, we can say that the movie Fantastic Beasts and Where to Find Them is at least a soft prequel, given that it takes place in the same universe as the Harry Potter movies, and certainly contains events that not only precede them, but are even referenced in Harry Potter. Not only that, the events in the Fantastic saga (pentalogy) will no doubt influence and shape events and characters in Harry Potter, specifically Albus Dumbledore.

That being said, we can't say that it is stricly speaking a hard prequel, because the events that happen throughout the film, Fantastic Beasts and Where to Find Them, do not directly lead up to the first chapter of the Harry Potter series, so the most we can say is that it is a soft prequel, since it takes place in the same universe, before events of the existing work.