Difficult this as I can think of more music that makes me cry, rather than songs. But there are two.
When I saw the film Watership Down, Bright Eyes but I was only young then.
Back to Black by Amy Winehouse as I always think it was such a waste of a good talent and it was a really honest song from her about her demons.
Another one I have thought of, Have Yourself a Merry Little Christmas by Judy Garland from the film Meet Me in St Louis. I always used to watch it with my mum and now that song starts me off.