Edward is Connors grandfather...I've been playing it 5-10 hours and thus far it has extremely little to do with the Assassins and Templars and the modern day story could very easily have nothing at all to do with the AC storyline.
That's something that I'm really wondering about myself. Edward Kenway is obviously a relative of Desmond, which is why we're getting his memories. But I'm trying to figure out how this is possible.
Altair led to Ezio, who were distant relatives of Desmond. But according to this new story, Edward just happened upon an Assassin and stole his clothes. I find it extremely coincidental that Edward would share lineage with Desmond's line of Assassins.
AC3 had almost too much exposition explaining Connor and his family history. This game has almost no exposition, and I'm wondering when Edward's story is going to fit into the greater Assassin's Creed plot.
Haytham is Edwards son...
Altair, Ezio, Aveline and the Kenways are different branches in Desmonds family tree that acummulated in Desmond...
There is an in-game e-mail which may be a teaser/easter-egg from Ubisoft Montreal about other branches/ancestors of Desmond based on his processed DNA:
Spoilers obviously
http://kotaku.com/assassins-creeds-wild-fake-email-in-joke-or-tease-fo-1456984380