I’d guess it would be Godspeed You Black Emperor!'s F# A# Infinity. However, all their stuff is somewhat thematic, so if that counts, so does all of their stuff.
Others would be Nine Inch Nail’s The Downward Spiral and (I would argue)the Beastie Boy’s Ill Communication*
Eminem’s albums continually tell the story of his alter-ego Slim Shady’s exploits. Love it. I would have to say “The Slim Shady LP” is my favorite. I never knew how many times a guy could murder his wife . . .
“Lola vs. Powerman and the Moneygoround” by the Kinks. Extra points because, for purposes of the album (its about the record business), the artist involved had a hit single – and “Lola” is that song.
Pink Floyd yet again with The Final Cut. I used to hate it, or at the very least find it pretty mediocre. Now I love. I don’t mind later day Floyd, but they were definetly Roger Water’s band.
The Beatles – “Sergeant Pepper’s Lonely Hearts Club Band”
The Who – “Tommy,” of course, but also “Quadrophrenia.” Someone once argued that “The Who by Numbers” was also a concept album, but I don’t buy that. “The Who Sell Out” is an very early concept album, too.
Van Dyke Parks – “Song Cycle”
Amazon.com has an interesting page listing concept albums. They’re right about “Pet Sounds” and I’m delighted they thought of “12 Dreams of Dr. Sardonicus.”
Tells the story of the band Omega and The Mechanical Animals, in their self-destructive rock opera drug orgy glory. Omega has an obsession with a girl/drug named Coma White.