Now that the movie has been released, we can complement KutuluMike's answer.
Tl;dr : the new Fantastic Beasts saga is more a
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,