Basically, songs that are more than just something to carelessly sing along to. Songs to, I don't know... evoke strong emotional reactions in you. Good or bad. You get the point, right?

For utter morbidity and shame: Billie Holiday - Strange Fruit
For heartfelt emotion and amazing voice: Antony & The Johnsons - Cripple and the Starfish
For great sense of atmosphere: Anna von Hausswolff - Gloomy Sunday

You go!