Adam, the following are the songs recommended by people. Mind that there are no particular order in this list.

1. 백만송이 장미 (One Million Roses) by 심수봉 (Shim Soo-bong)
http://www.youtube.com/watch?v=_D4V_qfTQq0

-A Wikipedia article on the song (in Korean)
http://ko.wikipedia.org/wiki/%EB%B0%B1%EB%A7%8C%EC%86%A1%EC%9D%B4_%EC%9E%A5%EB%AF%B8

-Interesting article on what’s behind the song, written by Royce Ng. (in English)
http://onemillionroses.blogspot.com/2011/03/sim-soo-bong-was-female-singer-from.html

-The original version of the song by Alla Pugacheva
http://www.youtube.com/watch?v=oIFmhye6fqw

2.Zigeunerweisen (Gypsy Airs), Op. 20
http://www.youtube.com/watch?v=_W_H2BEJbt0

Wikipedia entry “Zigeunerweisen” http://en.wikipedia.org/wiki/Zigeunerweisen (in English)

Its popular appropriation was for a famous comedy actress 이영자(Lee Young-ja), who is in this video clip (click here) during the 1990s. Whenever she cried to say something to her fat, Zigeunerweisen was played loud as a background music.

3. “The Unforgiven” by Metallica
http://www.youtube.com/watch?v=TUHFfR8hWcA

4. “Hide and Seek” by Imogen Heap

5. “돌이킬 수 없는 걸음 (Irreversible)” by 이병우 (Lee Byeong-woo) from the O.S.T of “장화, 홍련(A Tale of Two Sisters)”
http://www.youtube.com/watch?v=sUbt6Hd5cug

Wikipedia entry on “Tale of Two Sisters” http://en.wikipedia.org/wiki/A_Tale_of_Two_Sisters and a trailer of the film http://www.youtube.com/watch?v=anF5XiN8QY8

6. “Lost in Your Eyes” by Debbie Gibson (1988)

Wikipedia entry on the song: http://en.wikipedia.org/wiki/Lost_in_Your_Eyes
and the lyrics: http://www.romantic-lyrics.com/ll9.shtml

7. 아빠와 크레파스 (Papa and Kurepasu(Pastel Crayon))

It’s a Korean song for kids, but the peculiar gloominess of the song made this song popular for almost every person raised in Korea. Have a look at the following two versions of the song:

http://www.youtube.com/watch?v=scz5rmgjH6A

Here is link for the lyrics: link

It’s unfortunately in Korean, so I’ll provide you a terse English translation of the lyrics.

어젯밤에 우리 아빠가 다정하신 모습으로 My father came back home last night smiling
한손에는 크레파스를 사가지고 오셨어요 음음 He had Kurepasu in his hands, Mm Mm.
그릴 것은 너무 많은데 하얀 종이가 너무 작아서 There were too many things to draw but the white paper was too small
아빠 얼굴 그리고 나니 잠이 들고 말았어요 음음 I fell asleep after drawing my father’s face Mm Mm
밤새 꿈나라에 아기 코끼리가 춤을 추었고 In the dream land, baby elephant danced all night,
크레파스 병정들은 나뭇잎을 타고 놀았죠 음음 Kurepasu soldiers played riding leaves Mm Mm
어젯밤엔 달빛도 아빠의 웃음처럼 나의 창에 기대어 Last night, moon light was on my window like my father’s smile
포근히 날 재워 줬어요 음음 Letting me soundly sleep Mm Mm

8. “상처만 (Scar)” by BOIS from the original soundtrack of “시크릿 가든(Secret Garden”

The intro of this song is used in a comedy show.

Song:
http://www.youtube.com/watch?v=jexSv6wJg84

Comedy show:
http://www.youtube.com/watch?v=StD4AhL0yFk

9. “Fire and Rain”(1970) by James Taylor

The song is featured in a film “Running on Empty”(1988) in the following scene:

There are many theories about the origin of this song, one of them being that the song was written after the singer’s fiancee passed away. See Wikipedia entry on “Fire and Rain” http://en.wikipedia.org/wiki/Fire_and_Rain

10. “Why do I need feet when I have wings to fly?” by Lucid Fall
http://www.youtube.com/watch?v=Ag_3XFmE0q

The title is a sentence said by Frida Khalo, after she lost her legs. You can hear the original version on this blog post: http://blog.naver.com/PostView.nhn?blogId=klklkl18&logNo=130109607083

11. “우리들의 죽음 (Our Death)”(1992) by 정태춘 (Jung Tae-choon)

The narration at the start of this song tells a story of two children who were burnt to death, while their parents went out to make money with the door of their small flat locked. The song was first released in 1990, but was censored. It was 1992 when the song was officially released, included in an album titled “아, 대한민국 (Ah, Korea)”

Here is a link for a blog article about the singer, Jung Tae-choon (in English): http://kocpop.blogspot.com/2011/03/jeong-tae-chun-and-park-eun-ok-92-rainy.html

12. “무시로 (Anytime)”(1989) by 나훈아(Na Hoon-a)

http://www.youtube.com/watch?v=j168um252W

It’s a sad love song about lamenting the ended love, crying all the time.

13. 상여소리 (Music for Death Carriages)

http://www.youtube.com/watch?v=PSf6GoLl99I

“상여(sang-yeo. meaning Death Carriage)” is a carriage for the dead. It is used during the funeral, with people (usually family members and relatives) carrying it to the grave site. Click the following link for an image of “상여” –> click

The below video is a very eloquent version of this music.