I hate to go all non pop (song) on this thread, which is the original intention, but the 60's was a great decade for film music too.
Bernard Herrmann was one of the main contributors to this decade, as well as one of it's biggest influences on other composers.
There were so many, but here's two:
Psycho:
http://www.youtube.com/watch?v=qMTrV...C4CC0&index=10
Cape Fear:
http://www.youtube.com/watch?v=QH3RhbVP7cA
I hope this isn't too far OT.
(I do not qualify these as songs, but I could make the argument that they are, if I had to :-)