Obviously, massive *** SPOILERS ***
1) Who is Rey, and where does she come from? To me, it seemed during the movie she would be revealed to be either Han and Leia's daughter, or Luke's daughter. I think the chances of her being Han and Leia's are pretty slim now, but there's still the possibility of Luke. From the intense Force-flashbacks she had after touching the lightsaber, to her quick-learned fighting ability, this would seem to be a logical possibility. But daughter by whom? Or is she a completely independent person who happens to be Force-sensitive, and will likely be part of Luke's new Jedi order.
But the other half to this question is why was she on Jakku? She apparently was either stolen or left there at a young age, with a promise that someone would be back for her. Who was it? And why was that promise or mission so strong she felt so compelled to return there time and time again? What is her purpose in the greater story?
2) What happened to Kylo Ren to turn him to the Dark Side? All we really know is he was Han and Leia's son, Luke tried to train him, he turned to the Dark Side and Luke ran away. There's a huge untold story there, and I'm not sure how much we'll be told in detail (at least within the framework of the actual trilogy). He seemed very young to have risen to such a position in the First Order, and immature as well. Obviously he has a volatile temper and is easily provoked. What components of his character led him to reject his parents and mentor so utterly and go down a path that would lead to Han's murder?
The other half of this question is what exactly happened to Han and Leia that they split up, and were estranged from each other? Was it Kylo's betrayal that caused their rift, or was it the reverse - their disagreements led to Kylo's feelings of abandonment and anger? Now that Han is gone we may never get the full story, and I'm not sure of Leia's involvement in the ongoing story. The answer to this question is likely a key aspect to Kylo's character so I believe it must eventually be addressed.
3) Who is Supreme Leader Snoke? What is he? Is he a Force-sensitive as it was hinted? Where did he come from, how did he get to be Supreme Leader of the First Order? Is he a Sith Lord, and Kylo is his apprentice? Or is he something completely different that will be introduced into the mythology? Does he really look like the creature that was in the hologram, or is there someone else "behind the curtain"?
Another part of this, is what exactly does "First Order" mean? It's such an arbitrary phrase I think it has to refer to something specific. How did it come to power on the ashes of the Empire? I would guess there will eventually be books and other supplemental material to address this, but I hope they don't leave it so ambiguous in the next movies that it becomes a mystery that won't be solved. How did it become so powerful that it could build a "Death Planet" without the New Republic even knowing about it until it was operational? How much of the galaxy does it control? How much does the New Republic control? And if the New Republic is the legitimate government, why are their defenders called The Resistance? Typically Resistance is synonymous with Rebellion, and the Rebel Alliance was "resisting" the established Empire. Why is The Resistance resisting an outside force?
4) How did Han and Chewie really lose the Falcon? I know there was a story how it was stolen by this guy, and that guy, and that guy, and eventually ended up on Jakku. But depending on what connection Rey is to the larger mythology, that can't be a coincidence. I think it was left deliberately there on Jakku for her to eventually use to escape. But it seems hard to believe Han and Chewie would abandon their beloved ship without good reason.
For that matter, how did they happen to be near the Jakku system just at the moment Rey and Finn needed rescuing? I think Han mentioned something about this in the movie, but it seemed a very convenient explanation.
Again, what happened between Han and Leia to cause his self-imposed exile for so many years, going back to smuggling apparently but without the Falcon? Hopefully this story isn't what it seems at face value.
5) Why is Luke really in exile? What happened during the process of beginning to train a new Order of Jedi Knights? Where are these other trainees? Did Kylo kill them, similar to how Anakin and the Emperor purged the Jedi Knights, or have the scattered to the winds? It seems out of Luke's character to run away and hide. I get the feeling there is a lot more to this story than simple shame and guilt that would drive our heroic Jedi to flee. If so, why leave behind parts of a map with R2 and Lor San Tekka/Poe Damaron/BB-8 that would help the Resistance find him? I think his disappearance is part of a much larger plan that he has put together to either bring down the First Order, recover Kylo Ren to the Light Side, or train Rey in the ways of the Force. Or there may be, and I hope this is the case, a stronger cause to create a new Order of Jedi that is nothing like what was there before.
Oh, and who was Max Von Sydow's character Lor San Tekka, and what was his connection to Luke? He was there for so short a time to deliver the map portion to Poe we don't know anything about him. He obviously had some kind of history, but he's dead so who knows if his character will be referred to again. Why is Poe Damaron such a good pilot? Is he Force-Sensitive as well? His convenient disappearance/reappearance seemed very pat - there seems to be more to his story. And while we're at it, how was Finn such a good lightsaber fighter out of the gate, without even knowing anything about the weapon? And how was he so lucky, and able to break his conditioning? Are they all Force-sensitives?
Those are my five burning questions (ok, with lots of corollaries). What do you think?