Note: I initially read this as "favorite albums" and posted as such. I deleted my last post because I didn't think it was really representative of "greatest albums." Apologies if there are still remnants of my replies to the nonexistent "favorite albums" topic in these entries.
Days of Future Past by The Moody Blues
THE symphonic rock masterpiece. It deserves every bit of popularity it has, critics be damned. This account of a day in the life of an everyman is as musically and lyrically inspired as you're likely to hear from any album. Its two most popular songs are also its best, but the album MUST be heard in its entirety for full effect.
Best Songs:
Tuesday Afternoon,
Nights in White Satin
In the Court of the Crimson King by King Crimson
An immortal classic. 21st Century Schizoid Man shook the rock world, but really every song on it, even the cryptic and (in my opinion) fairly boring Moonchild, is a force to be reckoned with.
By the way, the iconic screaming face on this album's cover is a pretty good representation of the atmosphere here, and what your reaction ought to be to this music.
Best songs:
21st Century Schizoid Man,
In the Court of the Crimson King
Leftoverture by Kansas
Kansas' most important album. This one has Carry on Wayward Son on it, but also all of the prog elements that made its predecessors great.
Best songs:
Carry on Wayward Son,
The Wall
Fragile by Yes
If you want to know what was great about 70s rock, this might be a good place to start. This is early Yes' most accessible, but also one of their most well-composed, albums.
Best songs:
South Side of the Sky,
Roundabout
2112 by Rush
Every song here is good, but it's the title track that steals the show. I also like the ballad Tears, which is unusually sentimental and unlike anything else Rush ever played.
Best songs:
2112,
Tears
Pet Sounds by The Beach Boys
No "greatest albums" list would be complete without it, and I'm surprised not to see it on everyone's list. What was done here for a unified sound, production, and composition was groundbreaking.
Best songs:
Sloop John B,
God Only Knows
Seventh Son of a Seventh Son by Iron Maiden
Oooh boy. This is easily one of the best concept albums ever, and one of the best albums in metal. The concept's loose, but it's so well-presented, and the music is so good, that there's really no way it can go wrong. Iron Maiden's best, and that's saying something.
Best Songs:
The Evil That Men Do,
Seventh Son of a Seventh Son
Images & Words by Dream Theater
Not enough good words can be said about this album. All the right elements are in place. Each song flows seamlessly into the next, and there's this magical atmosphere of wonder throughout. The lyrics can be "word salad," but they're also mystifying and poetic and serve the music well. This has the perfect balance between prog and metal as well, but is packed with rich, accessible melodies that stay with you. I believe this album is genuinely flawless.
Best songs:
Under a Glass Moon,
Learning to Live
Red by King Crimson
Another massive success by King Crimson. Influential in grunge and probably metal.
Best songs:
Red,
Starless
V by Spock's Beard
Yeah, you might laugh when you first hear this band's name. The poppy vocals might merit another chuckle, too--but make no mistake, this is a band to take very, very seriously, and the two monster tracks that bookend this album prove it. The four tracks in between aren't filler, and are also diverse enough to make each listening experience interesting. This is the band that really got me into prog, and you have to hear it to know why. "The Great Nothing" alone, in spite of being nearly half an hour long, is so brilliantly written that it's hard not to hit that repeat button when the last poignant note plays.
This really is the culmination of everything great about prog rock. It may not be influential, but it's great.
Best songs:
At the End of the Day,
The Great nothing
Honorable mentions:
Sgt. Pepper's Lonely Hearts Club Band. Yeah, it pretty much needs to be on this list.
Tommy by The Who
Selling England by the Pound by Genesis
Close to the Edge by Yes
Black Sabbath by Black Sabbath
Also, any album by Simon & Garfunkel.