Naturally, any type of program music of the "narrative" variety, although its quality does not necessarily suffer if you have not read any program notes.
This trope applies to a lot of classical music. In twentieth-century and contemporary works especially, there may be a lot of details in the score, special performance notes, and analytical information that is known only to the performer or to a scholarly audience.
In the instance of Charles Ives's Concord Sonata - a musical depiction of the transcendentalist movement in Massachussets - a pianist may have read selected writings of Emerson, Hawthorne, Bronson and Louisa May Alcott, and Thoreau, as well as Ives's own "Essays Before A Sonata" (an extended program note on the sonata), performance directions, and numerous supplemental materials, and still have no clue as to how to interpret the work except by intuition.
The Second Viennese School of Arnold Schoenberg, Anton Webern, Alban Berg, and their subsequent influence turned out to be the apex of this in the early twentieth century. The techniques and aesthetics employed, including twelve-tone rows, athematicism, and non-triad-based harmony, had placed such composers at the furthest reach of comprehensibility while still maintaining a system that was, at least internally, organized by some logical system. You have to know the score well, in other words.
Parts of the backstory for the Rock OperaSpace Crackers can only be found on the band's website.
Marilyn Manson's Triptych, consisting of the albums Antichrist Superstar, Mechanical Animals and Holy Wood takes this and runs with it. Reading the actual booklet that comes with each album helps, since each of them are concept albums, and it gives you the name of the protagonist of Antichrist Superstar (all three of his names, actually), the names of the protagonists of Mechanical Animals as well as which of them sings which songs, and some stuff for Holy Wood, although not much. This is because there was supposed to be a Holy Wood novel, and Manson still wants to release it as of mid-2015. Holy Wood came out in 2000. The interviews help a bunch more, giving details like the fact that the Triptych story takes place in the opposite order of release. Holy Wood and Mechanical Animals, therefore, are prequels, which makes sense, since reality is destroyed at the end of Antichrist Superstar. Perhaps the most helpful source, however, isn't even an official one. The website "The Nachtkabarett" has over a hundred essays explaining all the imagery used in the art, costumes, stage shows, lyrics, symbols and other stuff, as well as deconstructing the lyrics, terms and so much more. It's pretty much the only way you're going to be able to understand it all, and it's taken many years to put together. It does the same for all the other albums, but in comparison to the Triptych, they're far more simple (emphasis on "in comparison to"). It says something that, with the first/last album in the series, Antichrist Superstar, having its 20th anniversary in 2016, not everything is known. For example, is "The Worm" (the protagonist of Antichrist Superstar) the son of Adam Kadmon (the protagonist of Holy Wood) and Coma White (his love interest, as well as Alpha and Omēga, the protagonists of Mechanical Animals) or not? Are Alpha and Omēga alive in Antichrist Superstar''? What's the significance of the February 14th, 1997 show (an in-universe concert, since Alpha, Omēga and Adam are all musicians), and which protagonist is playing at it (there's one song on each album from it, which are either live or pseudo-live)? This is a case of All There in the Manual where the "manual" is interviews, music videos, a fansite, booklets, and an unreleased book (besides for Chapter 10). Until the book is released, and likely far after that even if it ever is, it's going to be a puzzle, and that was intentional, so answers will be slim.
A less painful example is Manson confirming that The Pale Emperor is based on the legend of Faust, with himself as Faust.
All of Coheed and Cambria's albums are about The Amory Wars, a sci-fi story by frontman Claudio Sanchez. The only way to truly understand the music is to read the comics, which so far have only covered about 2/5 of the saga. The rest is pretty much guesswork.
Without the stories that accompany The Residents' album Eskimo, all you'll be hearing are wind noises, tearing, and grunts.
Though you can understand the songs on their album Animal Lover, the manual does help a bit.
Avantasia suffers from this. Sure, the music may make some coherent sense listened to by itself, but to get the entire story, you need to read the liner notes.
Also, Blue Oyster Cult's Imaginos suffers from this. Doesn't help that they were in a botched track order by the record label...
The Who wrote two musical Rock Operas, Tommy and Quadrophenia, as both albums and stage musicals. The stories of both of these albums can be difficult to figure out using just the music and lyrics, watching the film versions and reading various analysis of the albums is required for the full story.
The same can be said of the mini-opera "Wire and Glass," though the originator of the kind, "A Quick One While He's Away," is blissfully self-explanatory.
Really, any rock opera qualifies. Including a lot of things by Trans-Siberian Orchestra. While the songs usually make sense in their own context, the backstory to them is usually found online or in the song booklet.
The albums of The Protomen are packaged with liner notes containing pages of narration and stage direction which are not included in the music proper.
It is not too hard to grasp the story, although details will be lost without the notes. The first track of the first album begins with a narrator telling of Dr. Light building Protoman to fight an oppressive robot empire led by Wily. Proto Man dying is clearly told. Then comes an instrumental, followed by Light arguing with Mega Man about fighting and why he believes it is useless. The next two songs are of Mega Man deciding to fight and his lust for vengeance. Finally Face–Heel Turn Protoman's speech and their battle. The second album can clearly be conveyed to be of first "Tom" being betrayed by his friend while building a mechanical work force, then being framed for murder, then a youth fleeing the oppressed city, then the two teaming up and trying to start a revolution. As said, things like why Light built Mega Man and how the "revolution" occurred can only be understood from the notes.
Michael Nesmith's 1974 album The Prison was famously marketed as "a book with a soundtrack." That is, the album was packaged with an actual novella (written by Nesmith), and you're supposed to read it as you listen to the album. 20 years later he tried that format again with The Garden.
While Todd Rundgren's Something/Anything is perfectly listenable on its own, Rundgren's liner notes for the album put the songs into a more coherent context. Each side (it was originally a double vinyl album) had its own thematic concept, and on the last side, Rundgren turned an eclectic group of songs into an "operetta" via a clever liner-note narrative that linked the songs together.
Pearl Jam's "Jeremy" from Ten is about a kid who kills himself in front of the rest of his class. You wouldn't know this unless you saw the music video. And even if you did, it was more than likely the one that was edited to remove a key shot vital in making it even remotely obvious.
Roger Waters' Radio K.A.O.S. Try linking the songs without knowing the story.
Even if you do know the story, that still doesn't mean it makes any sense at all.
The mysteries presented for listeners in Nox Arcana CDs are near impossible to figure out unless you visit The Arcanum on their website. And even then, some of the puzzles (The Doctor's Office and Pirate Treasure come to mind) will still require visiting the "Hints" section of the forum. In a much less psychologically challenging vein, all of the stories in the songs are much better understood once you've read the full stories/poems.
The CDs of Standing Stone, a mostly-instrumental classical work by Paul McCartney, includes a poem that includes the storyline for the piece. The NPR broadcast from when it was released did not.
David Bowie's Rock Opera1. Outside has a short story in the liner notes setting up its storyline and major characters. Unfortunately, this only takes the story so far because the album was intended as the first of three; Bowie decided not to write/record the follow-ups, so the world will probably never know how everything was going to work out.
Pepe Deluxé's album Queen of the Wave is a Rock Opera involving an epic hero and the fall of Atlantis, based on the novel A Dweller On Two Planets. To pick up anything beyond the broadest strokes of the story (like, say, the characters' names), you need to read the album liner notes, or the "album companion" files that the band posted online. PD themselves said that they were more concerned about making the album an enjoyable listen on its own than they were about conveying the entire plot via lyrics.
The album Ghosts and Spirits is based heavily on C.S. Lewis' book The Great Divorce. Considering the nature ofthe book, understanding the references, conversations, allusions, context, and even the title is virtually impossible if you aren't familiar with The Great Divorce.
Garth Brooks' In the Life of Chris Gaines, presented as a Greatest Hits Album, describes Chris' life in the liner notes: he was born in Australia to an Olympic swim coach and swimmer, but moved to America and was raised there until he dropped out of high school to form his band Crush, which scored their only hit song, "My Love Tells Me So". After one of his band members died in an airplane accident, Chris Gaines went solo with his first album Straight Jacket in 1989, followed by Fornucopia in 1991, Apostle in 1994, and Triangle in 1996.
Moby's Play contains short essays about veganism, religion, and politics. These are pretty Anvilicious and would get a lot of people to stop listening to his music outright, but Moby explains at the end that these essays serve to explain his motivation to write lyrics in the first place. He closes them by saying, "You might hate the music, but you might love the essays. Who knows, by some miracle, you'll like them both."
Rhapsody of Fire's albums all tell a high fantasy story, but to actually understand the plot you need to read a booklet in a physical copy of the album.