I just played American Idiot and The Wall from start to finish. For me, both are absolute masterpieces. I wouldn’t be able to say which is better than the other.
I've listened to most of the other concept albums mentioned in this thread, but I believe deep in my heart of hearts, this album has the strongest story and emotional impact.
21st Century Breakdown also does a good job, it's a little different but it's got the same album story.
Deltron 3030 is a stellar album about an intergalactic rapper. It's produced by Del tha Funkee Homosapien and Dan the Automator, seriously, killer.
And before the Flobots came, there was Johnny 5 and Yak who made Onomatopoeia, another stellar album. It's an underground 2000's set though, so it gets gritty. But it's very real and so well composed. That of course led to Flobots Fight with Tools, which may just be the best political album of all time.
SFaM was my answer. I bought that on CD when it was still newish and it was on repeat for a long long time. And I've listened to it at least twice a year for the last 15 years. (sobs) I'm so old.
If I die tomorrow
I'd be alright
Because I believe
That after we're gone
The spirit carries on
Great suggestion, but I'd actually say The Ballad of Dood and Juanita and Sailor's Guide to Earth are better concept albums from Sturgill Simpson. Although, OP seems to lean toward rock more than bluegrass/country, so Sound and Fury might be a better place to start.
It's a concept album by a folk-punk/cider-punk band about WWI, released for the centennial. True to the punk roots, most of the songs deal with the dark, ugly, and gritty sides of war.
There's "Anna Maria", a metal love/revenge ballad about a sailor losing his beloved brigantine to a U-boat.
"Gavrilo" about the horrific consequences of the murder of Archduke Ferdinand.
"Black Letters", a rather haunting song inspired by the last correspondence that a young Canadian soldier wrote to his wife.
"Back Home in Bristol" tells the story of a young man, afflicted by PTSD, facing court-martial for failing to obey orders due to his condition.
"A Broken World" is a reading of English poet Amy Lowell's "September, 1918", which is both heart-wrenching and blossoming with hope.
It's not all bleak though. There's also an original instrumental piece called "The Amiens Polka". And am original shanty called "The Bay of Suvla", which I feel is an upbeat prequel to The Pogues' "And the Band Played Waltzing Matilda".
To top it off, they saved songs that didn't make the cut and released a B sides album as well.
In my opinion, this album does such a great job of striking the emotions that it makes "The Wall" and any of the other "greats" seem vapid and mediocre.
But that's ok, different people like different things.
<3 That's very fair.
I personally find most of the "greats" to be significantly overrated. They were incredible for their time as a departure from what preceded them but, over-produced and lacking in genuine-ness. They may have had higher levels of instrumental virtuosity but lack relatability, depth, and exploration of the human condition.
To be fair, there's a lot of people who don't want any of that. Many people want escapism, heavy bass with questionable lyrics, or something weird and fun to trip balls to at a laser show. That's fine. I don't care for out of touch celebrity musicians; I want something "real" that evokes emotions across the spectrum and unafraid to dwelling in unpleasant places when the subject matter calls for it.
It's an absolutely amazing album but also the most tragically sad work I've ever engaged with in any medium. I've only listened to it once because it turned me into a complete mess. I don't know whether I'll ever hear it again.
Mount Eerie is Phil Elverum and A Crow Looked at Me chronicles the illness, death and grieving for his wife, artist and musician Geneviève Castrée. She was diagnosed with pancreatic cancer four months after the birth of their daughter and passed a year later.
It's filled with large and small observations, like the fact that there's no deeper meaning to Castrée's death and describing the things she did for their daughter before her passing, like buying her a backpack she knew she would never live to see her use.
If you're into indie rock, I highly recommend Separation Sunday by The Hold Steady and The Monitor by Titus Andronicus.
They're both very high-brow (or pretentious, depending on who you ask), but for my money the musicianship and the lyrics just keep me coming back.
Separation Sunday is all about growing up and doing sketchy stuff, finding/losing the faith you grew up with, and finding/losing friends to death, jail, and life.
The Monitor uses the story of an ironclad from the American Civil War as a metaphor for the internal struggle of a 19yo trying to leave his blue-collar roots behind and go to an Ivy League school. If you've ever felt that internal divide between who you are and who you want to be, it's a fantastic album. Shades of Bruce Springsteen with some modern punk.
Speaking of Springsteen, Born to Run could be considered a concept album and it's a classic, especially if you've ever felt the bone-deep need to get away from everything about your home town.
There are a lot of other great suggestions in this thread (especially Good Kid, MAAD City), but they're mostly the standard "best 50 concept albums of all time" that you see in lots of music journalism, so I wanted to make sure my favorites got seen, too.
Ooh, 'best' is hard. I think I'd have to go with Operation Mindcrime by Queensrÿche because I only ever want to listen to it as a complete album, start to finish.
Runners up:
Dark Side of the Moon - Pink Floyd
Seventh Son of a Seventh Son - Iron Maiden
Misplaced Childhood - Marillion
The Rise and Fall of Ziggy Stardust and the Spiders from Mars - David Bowie
Elizium - Fields of the Nephilim
and, if it counts, Murder Ballads - Nick Cave and the Bad Seeds