Scenes from an Italian Restaurant, Billy Joel.
It’s All Coming Back To Me Now, Celine Dion
The Boxer, Simon and Garfunkel
No Son of Mine, Phil Collins
Agree with almost everything by Loreena.
Another vote for this, apparently as a toddler I’d rock along to this on my rocking horse for the whole 7 minutes or so. Great lyrics and a good story.
Copperhead Row is a great song.
I prefer Thin Lizzy’s cover of Whiskey in the Jar Looking a the wikipedia article this may be the oldest song mentioned thus far.
Fairytale of New York is another great one.
Another one which I’m surprised not to have seen mentioned - The Band’s The Night They Drove Old Dixie Down.
By Shel Silverstein, actually. At least if we understand “by” to mean “written by.”
Maragritaville by Jimmy Buffet (what the hell is wrong with you people?)
Also
Uneasy Rider by Charlie Daniels
Loads of Irish songs
Molly Malone as sung by the Dubliners
Don’t Go for the One by Gaelic Storm
Darcy’s Donkey by Gaelic Storm (a total blast of a song)
Go here to see my most favorite singer James Otto perform “In Color” written by him and Jamey Johnson and Lee Thomas Miller in front of a crowd that I was a part just over two years ago which in my opinion could not be more of a story type of song. 
God bless you and James always!!!
Holly
P.S. This song was named the 2009 CMA and ACM Song of the Year.
I love him so much and it makes me sad everything that he might have done. Lies is probably my all-time favorite song of any genre, both the writing and his beautiful delivery.
/Stan Rogers Sweet Heart
This song is amazing! Very well played, gigi.
Cold Missouri Waters by James Keelaghan
He doesn’t actually sing it, of course. Does it count as a song?
Missed the edit window:
This one does, though. Angie Baby, as performed by Helen Reddy.
Chris De Burgh has a number of these
From *Spanish Train and other Stories *–
Spanish Train
Patricia the Stripper
The Painter
From Crusader
Crusader
The Girl with April in Her Eyes
Mr. Bojangles
“Drink with the Living Dead” by Ghoultown.
Folsom Prison Blues - Johnny Cash
Frankie and Johnny - Johnny Cash
You Never Can Tell - Chuck Berry
Back In the USA - Chuck Berry
(Actually, most of Chick Berry’s songs)
The Battle of New Orleans - Johnny Horton
Sink the Bismark - Johnny Horton
Fernando - ABBA
Whiskey Lullaby - Alison Krauss
Alice’s Restaurant Massacre - Arlo Guthrie
The Weight - The Band
Jail House Rock - Elvis
In the Ghetto - Elvis
The Sloop John B. - Beach Boys
Brandy - Looking Glass
+1.
Also, The Year of the Cat, Al Stewart, though more for the music than the story. And I am in absolute awe that they could make a live version come out so well.
A deck of Cards by Tex Ritter
Big Riff and Mando, Jethro Tull
Pretty much anything by C. W. McCall: Convoy, Wolf Creek Pass, Classified, Roses for Mama, Black Bear Road, Crispy Critters, The Little Brown Sparrow and Me…um, yeah. Just go listen to just about anything of his. 
Lyle Lovett - *Church *and Lights of LA County
The Ballad of John and Yoko.
Harry Chapin has been mentioned, but not Mr. Tanner. It’s a real three-hanky weeper, too.