That’s what I came in to say. Even my atheist Dad likes this song. A lot.
Everybody’s Changing by Keane.
It always makes me a little weepy for some reason.
You don’t have to be part of a group to like something that came out of it. I’m an atheist and one of my favorite songs of all time is Amazing Grace, and one of my favorite books is the bible.
Beethoven - Piano Concerto no 4, 2nd movement
Many have associated this piece with Orpheus’s taming of the Furies, but there is one school of thought which believes it is an allegory for his own worsening deafness (the sinister unison strings, against which the solo piano struggles, and is finally fully engulfed at the end); the piece was written around the same time he first confides that he is losing his hearing in one of his letters. In that light, I find the piece not only beautiful, but soul-crushingly terrifying.
Wow that’s exactly what I was looking for. Absolutely beautiful!
Someone already mentioned God Only Knows. I agree. Someone wrote down a couple of songs by Nick Drake. I’d like to make a couple of contributions: Nick Drake’s Place To Be and Israel “Iz” Kaʻanoʻi Kamakawiwoʻole’s Over The Rainbow.
Huh, here I was thinking I’d get my ears boxed if I posted an Apocalyptica vid - but since you broke the ice, so to speak, my personal favourite evocative song is Quutamo. The strings in the intro get me, every time. (Though, forgive, but the audio quality in the song deserves to be listened to with a good headset, as the bass is important.)
Biz Markie’s You Got What I Need 
When I am trying to sleep I run *The Vilja Song *from Lehar’s *The Merry Widow *through my mind. Sometimes that helps.
I’ll nominate Diane Schurr’s Guess I’ll Hang My Tears Out to Dry. You can tell she is crying in the last few lines of the song.
I Burn For You by Sting
The closing credits song from the film Spirited Away is stunningly beautiful
You really need the chorus for the full experience. http://www.youtube.com/watch?v=BotBJVg0ECM
I fourth or fifth Ode to Joy. There’s a reason it’s such a famous, nearly cliched, melody… which is that it is just so simple and beautiful. The 4th movement of Beethoven’s 9th is the perfect marriage of simplicity and complexity, of beauty and bravura artistic mastery.
As for things written in the past 50 years, it’s hard for me to really accurately assess something I’ve still only recently become familiar with, but I can’t say enough good things about The Song of Purple Summer from Spring Awakening by Duncan Sheik.
“Superstar” by The Carpenters
Another vote for “I Will” (the original).
I’m partial to Vienna Teng’s “Lullabye for a Stormy Night” lately.
Sweet melody, touching lyrics, soothing delivery…
(Sorry, only link I found was a Doctor Who mashup! 
)
Redeemed by Charlotte Martin. I can’t find a good YouTube link, but it’s really a gorgeous song.
Poliahu written by Frank Kawaikapuokalani Hewett