There are many beliefs about Lilith. Many of them do not call her a fallen angel. Some say she was Adam's first wife, as in created before Eve, but she was too strong-willed so God banished her from the Garden of Eden. Some say she was Satan's wife, and birthed demons. Others will say that second tale was only used to portray her as evil rather than good.
There have been many discussions about Lilith on this site. Some are quite informative amidst the debate!
I'd suggest doing your own research and coming to your own conclusion. After all, I'm just some words posted on the internet. Why trust what I have to say?