Wow...this is tough. I love movies, all kinds.
Of course, if the criteria is "haunt you" then you have to go to the more "unpleasant" movies:
Distressing/moving/dramatic: Probably Sophie's Choice. Schindler's List is a close second. The subject matter alone is enough.
Horror Movies: Prince of Darkness. This is a 1987 movie that just creeped me the hell out. Still does if I think about it too much.
Now...if you just want "stay with you" movies:
Chick flicks: Under the Tuscan Sun. May well be my all-time favorite...certainly in the top 2-3. "Unbelievably good things can happen." (Plus, there was a time in my life when the thought of just leaving everything and starting over somewhere new where no one knew me really appealed to me. These days the Villa in Tuscany (with Jabber) really appeals to me.

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Romance: The Quiet Man. This was my mother's favorite movie and until Under the Tuscan Sun, I would have said it was mine, even if that's the only reason.
Comedies: Monty Python and the Holy Grail. I can quote pretty much the entire movie from heart. (So can Jabber.)
Animated: Am I being too trendy to say Frozen? I love that movie! Finally one that has the women doing something other than waiting for their prince to save them, the "true love" is between siblings, and there's the whole bit about "let it go".
I just realized there's no sci-fi on my list, which is weird because the vast majority of what I watch is sci-fi/fantasy/super hero stuff. But really, it kind of gets interchangeable after a while. Of those genres...probably Serenity. Firefly was such an awesome show and the movie was just as brilliantly written.