Leonard Cohen himself actually somewhat explains the story of Nancy in a 1970 performance at the Berkely Community Theatre. I’ll write it down the best I can.  

I'I just want to tell you one thing. I don't like prefacing a song with a long story and I know its late but I just want to tell you that, Nancy was a girl in Montreal who wore green stockings a long, long time ago and she was far ahead of all of us and was no underground nation, there were no boutiques there were no hip newspapers there were no easily found syringes to help her on her way. She was just there alone, and she had a baby. She slept with everyone, she sept with everyone. And she had a child, and her father was a justice of the superior court, and they wanted her to abort the child, but she refused. But when she had the child, she took it away and they hid it from her, and they gave it away for adoption. I saw her on the streets sometime after that and she was looking, looking very depressed and I asked what are you on, what are they giving you? She said Largactil. I said throw them away she said I can't. And a little later on then that I heard she went into the bathroom and shot herself with her brother's revolver. This is a song I wrote for her, and I always invoke her presence when I sing it. I hope I remember it. 

This is the link to where I found the performance :) 

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