I have seen few narrations and some fan made videos on the battle between Dumbledore and Grindelwald. There was some magic that killed Ariana when both Dumbledore and Grindelwald were duelling. What may have killed her? Does the magic bounce off?
She was killed by a rebounding spell, but nobody knows whose.
In Aberforth's words (emphasis mine):
“And there was an argument… and I pulled 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 – ”
The colour was draining from Aberforth's face as though he had suffered a mortal wound.
“ – 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.”
-- HP and the Deathly Hallows, Chapter 28: The Missing Mirror
In Albus's words (emphasis mine):
“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.”
"Oh, not death," said Dumbledore, in answer to Harry's questioning look. "Not what he could do to me magically. I knew that we were evenly matched, perhaps that I was a shade more skilful. 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.”
-- HP and the Deathly Hallows, Chapter 35: King's Cross
Neither of them go into great detail on exactly what happened, presumably because they don't really know themselves as everything was happening so fast. But presumably none of the three was actually targeting Ariana (the only one who might have was Grindelwald, and he probably had enough on his hands dealing with the two brothers), so the spell that hit her by accident must have been either poorly aimed or, perhaps more likely, rebounded off another spell or object. And if you doubt the latter is possible, recall the following scene in HP and the Goblet of Fire (emphasis mine):
For a split second, they looked into each other’s eyes, then, at exactly the same time, both acted.
"Furnunculus!" Harry yelled.
"Densaugeo!" screamed Malfoy.
Jets of light shot from both wands, hit each other in mid-air, and ricocheted off at angles – Harry’s hit Goyle in the face, and Malfoy’s hit Hermione. Goyle bellowed and put his hands to his nose, where great ugly boils were springing up – Hermione, whimpering in panic, was clutching her mouth.
EDIT: according to Pottermore, the above quote from Albus suggests that the killer was most likely to have been Grindelwald:
Quite who cast the spell that killed Ariana is unclear, although when Harry meets Albus Dumbledore in King’s Cross after he is nearly killed during the Battle of Hogwarts, the old Headmaster strongly hints that it was Grindelwald who shot the mortal curse.
The simple answer is no one knows exactly who killed Ariana.
Many people believe that it was Grindelwald due to his ability to perform dark magic and (Albus) Dumbledore's claims. Here I want to emphasize another theory that is possible and somewhat controversial to the existing claims.
There are 4 people involved in this fight; Aberforth, Albus, Ariana, and Gellert. Let's narrow this down to each person. Aberforth and Albus are most unlikely to kill her because they loved her so much(even Albus). IMO, both these brothers didn't even target her at all. There is a very small chance that they were the culprits, but the probability is very low.
Then the most suspicious one here is the Gellert. Yes, during the fight he(Gellert) used Cruciatus Curse on Aberforth. But remember, he didn’t try to kill him. Therefore, I think, even though Gellert could kill Ariana, he wouldn't attempt to kill her because at the end of the day he wanted to maintain a good relationship with Albus.
The final one involved in the fight is Ariana. Before getting into that, remember how Kendra Dumbledore died. Ariana couldn’t control her magic, and due to that, there was a magical explosion that leads to Kendra's death. I think, a similar thing happened during this fight. Ariana saw her favorite person(Aberforth) suffer from the Cruciatus Curse. Therefore, she was extremely angry with Grindelwald; unintentionally, Ariana performed some uncontrolled magic. Grindelwald might try to block this uncontrolled magic and because of that, the magic rebounded. I think that kind of rebounded magic caused Ariana's death.