This unnamed woman was a psychologist working at Connor Beverly Behavioral Medicine Center.
In 2008, this psychologist was assigned to evaluate a patient named Anna Milton.
During her last session with Anna, the psychologist inquired as to whether Anna, who appeared to be a dazed state, if she knew where she was and why. She explained to Anna that the woman had gone hysterical, and that it took four people to restrain her.
Anna said she was trying to warn people—everyone, about the Apocalypse, which she admitted was stupid since the psychologist would think she was "nuts", yet Anna insisted she was telling the truth. The psychologist said she was willing to hear Anna out.
Anna went on to explain about the demon Lilith's attempt to break the 66 seals to free Lucifer from Hell. She proclaimed that Lucifer was going to bring about the Apocalypse. Anna paused her explanation when she began hearing distant whispers, to the confusion of the psychologist. Anna resumed by explaining that no one knew what seal Lilith was going to break, because there was over 600 in total, and because of that, the angels were losing and everyone was going to die.
After this warning, the psychologist did not see Anna again since Anna had made an escape from the hospital after a demon, in the guise of an orderly, tried to capture her. Given the psychologist's expression throughout Anna's story, she clearly did not believe the woman.
Later that day, the psychologist was approached by Sam and Dean Winchester, who were asking about Anna. The woman insisted that she genuinely wanted the help the young woman. She also suspected that Anna had planned her escape. She admitted to Sam and Dean that she believed Anna was suffering from schizophrenia, and was overtaken by delusions.
She told the brothers about Anna's father Richard, and presumably informed them about Richard's location, which became their next destination. She requested the brothers to find and retrieve Anna, out of concern for someone like Anna to be out on her own where it's dangerous.