Essentially Thror goes to see Moria, and they get attacked by Goblins & Orcs, which is the "Battle of Azanulbizar" which is the event that you see in the film, where Azog kills Thror. (In the book Azog is slain here, and Azog's son Bolg is the one attempting to take revenge, but for the narrative story in the film, I can see why they kept him alive).
Essentially, because of the significant losses that the dwarves sustained during the battle, and because Dain had seen the Balrog "Durin's Bane" (which is rather famous for driving the dwarves out of their kingdom in Moria originally), they didn't attempt to reclaim Moria after that battle. It was barely a victory for the dwarves, and essentially each side got massacred. (This is also a big point about Thorin's father
Thrain, but just be aware that that article will reveal a number of things that you're going to see in the second film, like how Gandalf got the key from him, and some other goings ons).
It's AFTER the events of The Hobbit that Balin takes some of his kin and goes to take back Moria, which is why Gimli suggests it in LotR, and why he heavily mourns when they find Balin's tomb. (Another moment in Fellowship that's going to be much more emotionally hard hitting, after getting to know Balin so well in the Hobbit trilogy).