They didn’t go into details about what happened between the Dumbledore brothers in the movie (Harry said he had no interest in it — he trusted Albus). Aberforth only said Albus sacrificed many things on his quest for power and gave their Ariana everything but time.
In the book, there are many more details. Most are given by Dumbledore to Harry in their conversation in King's Cross station and especially Aberforth's conversation with the Trio in his pub. Short version:
When she was little, she was attacked by Muggles
As a result of the attack, she was unable to control her magic (couldn't use it but it exploded out of her)
Her father was sent to Azkaban for attacking the Muggles at fault in revenge
Her mother moved so she wouldn't have to be sent to St Mungo's
Aberforth was the best at controlling her spontaneous magic eruptions and calming her down.
One day Aberforth wasn't there and she accidentally killed her mother
Albus had to interrupt his academic career and go take care of her because he insisted Aberforth needed to finish school
While doing that caring, Albus hooked up with Gellert Grindelwald.
Aberforth showed up and told Albus that his plan to travel with Grindelwald (with Ariana in tow) was stupid.
When Albus, Aberforth and Gellert had a fight, Ariana was killed.
Dumbledore didn't know which of the 3 killed her and not knowing was the main reason he postponed confronting Grindelwald until 1945
More full info:
Aberforth's story (Deathly Hallows, Chapter 28, "The Missing Mirror")
“When my sister was six years old, she was attacked, by three Muggle boys. They’d seen her doing magic, spying through the back garden hedge: She was a kid, she couldn’t control it, no witch or wizard can at that age. What they saw, scared them, I expect. They forced their way through the hedge, and when she couldn’t show them the trick, they got a bit carried away trying to stop the little freak doing it.”
“It destroyed her, what they did: She was never right again. She wouldn’t use magic, but she couldn’t get rid of it; it turned inward and drove her mad, it exploded out of her when she couldn’t control it, and at times she was strange and dangerous. But mostly she was sweet and scared and harmless.”
“And my father went after the bastards that did it,” said Aberforth, “and attacked them. And they locked him up in Azkaban for it. He never said why he’d done it, because the Ministry had known what Ariana had become, she’d have been locked up in St. Mungo’s for good. They’d have seen her as a serious threat to the International Statute of Secrecy, unbalanced like she was, with magic exploding out of her at moments when she couldn’t keep it in any longer.”
“We had to keep her safe and quiet. We moved house, put it about she was ill, and my mother looked after her, and tried to keep her calm and happy.”
“I was her favourite,” he said, and as he said it, a grubby schoolboy seemed to look out through Aberforth’s wrinkles and wrangled beard. “Not Albus, he was always up in his bedroom when he was home, reading his books and counting his prizes, keeping up with his correspondence with the most notable magical names of the day,”
Aberforth succored. “He didn’t want to be bothered with her. She liked me best. I could get her to eat when she wouldn’t do it for my mother, I could calm her down, when she was in one of her rages, and when she was quiet, she used to help me feed the goats.”
“Then, when she was fourteen… See, I wasn’t there.” said Aberforth. “If I’d been there, I could have calmed her down. She had one of her rages, and my mother wasn’t as young as she was, and… it was an accident. Ariana couldn’t control it. But my mother was killed.”
“So that put paid to Albus’s trip round the world with little Doge. The pair of ‘em came home for my mother’s funeral and then Doge went off on his own, and Albus settled down as head of the family. Ha!”
“I’d have looked after her, I told him so, I didn’t care about school, I’d have stayed home and done it.
He told me I had to finish my education and he’d take over from my mother. Bit of a comedown for Mr. Brilliant, there’s no prizes for looking after your half-mad sister, stopping her blowing up the house every other day. But he did all right for a few weeks… till he came.“
“Grindelwald. And at last, my brother had an equal to talk to someone just as bright and talented he was. And looking after Ariana took a backseat then, while they were hatching all their plans for a new Wizarding order and looking for Hallows, and whatever else it was they were so interested in. Grand plans for the benefit of all Wizardkind, and if one young girl neglected, what did that matter, when Albus was working for the greater good?”
“But after a few weeks of it, I’d had enough, I had. It was nearly time for me to go back to Hogwarts, so I told ‘em, both of ‘em, face-to-face, like I am to you, now,” ... “I told him, you’d better give it up now. You can’t move her, she’s in no fit state, you can’t take her with you, wherever it is you’re planning to go, when you’re making your clever speeches, trying to whip yourselves up a following. He didn’t like that.” said Aberforth,
.... “Grindelwald didn’t like that at all. He got angry. ...“And there was an argument… and I pulled out my wand, and he pulled out his, and I had the Cruciatus Curse used on me by my brother’s best friend – and Albus was trying to stop him, and then all three of us were dueling, and the flashing lights and the bangs set her off, she couldn’t stand it – ”
“ – and I think she wanted to help, but she didn’t really know what she was doing, and I don’t know which of us did it, it could have been any of us – and she was dead.”
And then Dumbledore talking to Harry, in Chapter 35, King’s Cross
“You know the secret of my sister’s ill health, what those Muggles did, what she became. You know how my poor father sought revenge, and paid the price, died In Azkaban. You know how my mother gave up her own life to care for Ariana.”
... “So that, when my mother died, and I was left the responsibility of a damaged sister and a wayward brother, I returned to my village in anger and bitterness. Trapped and wasted, I thought! And then of course, he came….”
“Grindelwald. You cannot imagine how his ideas caught me, Harry, inflamed me.
... “And then… you know what happened. Reality returned in the form of my rough, unlettered, and infinitely more admirable brother. I did not want to hear the truths he shouted at me. I did not want to hear that I could not set forth and seek Hallows with a fragile and unstable sister in tow.”
“The argument became a fight. Grindelwald lost control. That which I had always sensed in him, though I pretended not to, now sprang into terrible being. And Ariana… after all my mother’s care and caution… lay dead upon the floor.”
... "They say he feared me, and perhaps he did, but less, I think, than I feared him.”
... It was the truth I feared. You see, I never knew which of us, in that last, horrific fight, had actually cast the curse that killed my sister. You may call me cowardly: You would be right, Harry. I dreaded beyond all things the knowledge that it had been I who brought about her death, not merely through my arrogance and stupidity, but that I actually struck the blow that snuffed out her life.”
“I think he knew it, I think he knew what frightened me. I delayed meeting him until finally, it would have been too shameful to resist any longer. People were dying and he seemed unstoppable, and I had to do what I could.”