Keynsham by the Bonzo Dog Band, which made several “Best Concept album” lists in the UK in the early 70s.
Willie Nelson - Redheaded Stranger (already nominated, but worthy of a second)
Bruce Springsteen - Nebraska
Kenny Rogers - Gideon. Probably his most widely-forgotten album, but it was my first introduction to concept albums. And it had some pretty good songs, including “Home to the Rock” and “Don’t Fall in Love with a Dreamer”, (with Kim Karnes) which was later a hit single.
Ian Tyson - Cowboyography. At the risk of over-advocating a personal favorite, (I’ve mentioned it several times in this sort of thread), but it really is a first-rate piece of work, a classic among modern country/folk albums.
Emmylou Harris - Ballad of Sally Rose
Yeah, that’s weird. And it is a great album too. I’ll throw in Sinatra’s Songs for Swinging Lovers. And to irritate the Floyd purists, The Final Cut. Also, Let It Be, originally to be titled “Get Back” which was as a concept to get back to their rocking roots.
Has anyone mentioned Brian Eno? How can we have missed that?
No real need to pick a single album as pretty his full back catalogue is a concept in progress. Perhaps “music for airports” is worth a specific mention.
For me it’s difficult to listen to ANY Pink Floyd album and say it’s NOT a concept album. Their songs just seem to flow together beautifully.
Marillion’s Misplaced Childhood probably wins the title for me, but a couple of favorites that have not yet been mentioned:
Titus Andronicus - The Monitor
Okkervil River - Black Sheep Boy
The Decemberists’ Hazards of Love is another great one that’s already been listed above.
I’ll go out on a limb and nominate Killers by Iron Maiden, from the criminally overlooked Paul Di’Anno era.
Thinking about this, it strikes me as odd that the original Black Sabbath never really did a concept album. Paranoid is close, especially when you consider that it was originally going to be titled War Pigs. But then they close it with “Fairies Wear Boots”, which doesn’t really fit in.
I’m glad someone has heard of it.
It should start a subcategory of its own - science fiction themed concept albums. There’s Hope by Klaatu, there’s the War of the Worlds, and there’s Clockwork Angels by Rush … there must be many others, but those are the ones I know best.
I’ll second Mastodon’s Blood Mountain, Zappa’s Joe’s Garage, and Dream Theater’s Scenes from a Memory. I’ll also throw in a few I haven’t seen mentioned yet:
Blind Guardian’s Nightfall in Middle Earth
Iced Earth’s Framing Armageddon
Ayreon’s 01011001 (sometimes called Y)
Also almost evrything by Ayreon. I say “almost” because Actual Fantasy isn’t a concept album.
Mercyful Fate - Don’t Break The Oath.
Is Sleep Has His House a concept album, too? Because that one’s a masterpiece.
Alice Cooper Goes To Hell would have made a great musical.
I have to nominate Dan Fogelberg’s The Innocent Age.
May as well nominate Abigail by King Diamond, then.
Another one I forgot:
Quantum Hack Code by Amogh Symphony
Pretty much all of his albums, especially solo.
Good one.
Also 666 by Aphrodite’s Child.
I love this album, but what’s the concept?
There’s actually a poster here named Politzania! Every time he posts, that song gets stuck in my head for half a day. 
Meanwhile, how did I not mention “In the Aeroplane Over the Sea” by Neutral Milk Hotel!? My favorite concept album ever.
It’s not an easy one to see, admittedly. I’d have said it was about the narrator struggling to deal with love in the face of addiction, then maturing and coming to terms with it in the end. But here’s what Pitchfork had to say:
Yes! If they had only done one album, it’s the one I’d pick. If they had only done one song, it would have to be Killers. (With Wrathchild a close second).