We don't know for certain why she was sent there, since her parents (presumably the ones who had her committed) are dead now. This is introduced toward the end of Twilight, when James and Bella meet at the ballet studio. He explains to her that Alice was another big hunt for him, but the vampire who watched over her changed her into a vampire so that James wouldn't be able to kill her.
Up until that point, Alice's earliest memories were as a vampire; she couldn't remember anything of her human life. We can infer to a certain extent that her talent for seeing the future might have been a reason that she was committed to the asylum in the first place, since we know that the vampires with special talents brought those talents with them from their former lives as humans. Perhaps Alice's parents looked at her talent as a sign of psychosis, and had her committed for it.
During New Moon, Alice searches for clues to her past, but she comes up with little new information.