I stole the diadem. I sought to make myself cleverer, more important than my mother. I ran away with it.
--Helena Ravenclaw (DH31)
The Grey Lady is the Ravenclaw House ghost; in life, she was Helena Ravenclaw, the daughter of Hogwarts founder Rowena Ravenclaw. Helena stole her mother’s diadem to make herself cleverer and fled to Albania. Her mother became deathly ill soon after and sent Helena’s long-time admirer to find her and bring her back. When she refuses to return, the Baron (now The Bloody Baron) killed her and committed suicide in remorse (DH31).
Other canon notes and references
Harry and Ron first encountered this tall ghost as she glided past in the corridor while out looking for the Mirror of Erised (PS12). Harry learned Helena's life story when he asked her for help finding the Diadem of Ravenclaw hidden somewhere in Hogwarts Castle. Helena revealed that she inadvertently helped Tom Riddle get the Diadem to use as a Horcrux (DH31). Harry found the Diadem of Ravenclaw in the Room of Requirement, and it was destroyed by Crabbe's cursed Fiendfyre (DH31).
Before the release of Deathly Hallows, the actress Nina Young, who played the Grey Lady in the first film, was told a few things about the character by J.K. Rowling herself and passed those on in a note to a young fan. The ghost was "a highly intellectual young lady" and a woman with strong scholarly or literary interests. "She never found true love as she never found a man up to her standards."
Her original name in JKR's notes was "The Whispering Lady" (Pm).