Albinoni's Adagio for Strings is the one I instantly thought of - it's quite a well known piece of music often used in films, tv, etc. Don't know if you'd call it classical but many of the 'songs' on The Piano soundtrack are quite haunting.
Barbers Adagio for Strings is lovely. Intermezzo from Cavalleria Rusticana by Mascagni is gorgeous (it's the music from the old Kleenex adverts -the one with the harp).
Pavane for a dead princess ("Pavane pur une Infante Defunte") is by Ravel. Also worth listening to: 2nd movement of Rachmaninov's 2nd Piano Concerto (used in "Brief Encounter"), and the slow movement from Mahler's 5th Symphony (used in "Death In Venice".)
i'll tell you doodlebug. I'm going to compile a cd for my father of classical songs, which represent as many moods as possible (e.g happiness, sadness, anger, love...) Sounds a little corny, huh? Thanks everyone for your suggestions