Gangsta’s Paradise comes to mind. Here’s a taste:

“They say I gotta learn, but nobody's here to teach me
If they can't understand it, how can they reach me?
I guess they can't, I guess they won't
I guess they front; that's why I know my life is out of luck, fool!“

Meg Myers "Monster" . I absolutely love this song, and the lyrics:

Actually, I sometimes don't know what a favorite song is actually about, I just like the beat or sound, but yes, other times you can't help but be struck by the lyrics. One for me is the Who's "Behind Blue Eyes"

Imagine by John Lennon

"I'm gonna find myself a girl
Who can show me what laughter means
And we'll fill in the missing colors
In each other's paint-by-number dreams
And then we'll put our dark glasses on
And we'll make love until our strength is gone
And when the morning light comes streaming in
We'll get up and do it again
Get it up again"

"Regression" by the Legendary Pink Dots - because i like the theatrical element to it:

And perhaps "You, Me, and Rainbows" from the same band for similar reasons:

The whole album Disentigration by the Cure.

Dark Globe by Syd Barret. Or classic Dylan. Or The Man Who Sold The World (Bowie). Or Nick Cave, Oh Children. Or Jennifer's Viel by the Birthday Party. Or . . .

Oh yes, forgot the why? Music, lyrics... an aphrodisiac for the romantic.

Hi again... I posted a comment yesterday, thought of a love song by Anne Murray....
"Could I Have This Dance"... schmaltzie... but says it all in not a lot of words.
... look it up...
Okay, so... "Could I Have This Dance" is my song with a story... not long ago... met a lovely lady on line... I was smart enough (?) to talk about romance, think about the song... and, being a gentleman, said the nice things a lady may like to hear. After several emails, telephone talks... now, I in live AZ... she, NC... she mentions spending christmas back home in OH... long story short... few other events made me think I should share my religious history, and my current non existent religious status. Y'all likely know, NC is the eastern capitol of the bible belt, so the forthcoming visit I (we) planned... canceled... broken hearted (15/20 minutes).
I would have used her name, don't like referring to her as she or her, so impersonal, oh well.
I don't know if that qualifies for a song that tells a story, but that one comes to mind... recalling a few more... think I could write a book...

Music moods vary for me daily, hourly, from Kris Krisstofferson's "Sunday Morning Coming Down"... hours of Jobim's Brazilian tempos, Johnny Hartman, Ella, Peggy Lee, Karrin Allyson, just to mention a few.
Music speaks to me, I listen to Pandora 10 to 11 hours every day, great variety.
I don't have a story to tell other than music has always been relatively important to me.

Lyrically, my favorite has got to be Bron Yr Aur by Led Zeppelin.

Express Yourself by Madonna. Women are taught to be quiet.

Bolivia by Gato Barbieri: The screating soprano sax and the tropical drums - haunting, or

Ain't No Sunshine, by Thelonius Monk off his Blackness album. Deep, slow sonorous voice.

"Ripple" by the Grateful Dead.