Fiction often works on several levels. On the one hand, you have the story that's actually being told, and the characters the story involves; on the other, you have the meaning behind the story, abstract ideas represented by the characters and symbols reflecting the story's view of the world. Even if the writer didn't intend for the story to have a deeper meaning, applicability may give it a resonance and context that goes beyond the basic plot. Moby-Dick is a story about an obsessed whaler hunting down a monstrous whale, but it's also about humanity railing uselessly against an uncaring universe. Lord of the Rings may have been intended just as a fantasy epic, but it has symbolic applications ranging from World War 2 to the loss of innocence.
But then sometimes you have stories with no literal level. At all.
These are the stories where the only answer to "why did the main character turn into a hawk and fly into a volcano at the end" is "because the hawk represents his desire for freedom and the volcano is the burden of free will". There is no straightforward, realistic storyline masking the symbolism, there's not even a Magic A Is Magic A sense of logic tying it all together. This is subtext without the text; subtext as text. The characters are complex self-representing archetypes and the events that unfold make no physical sense whatsoever unless you take the whole thing as an semiotic play of some kind.
Needless to say, this sort of plot can prove frustrating for literal-minded audiences who expect a more concrete answer to a plot hole than "because the plot hole symbolizes our inability to define existential truth". Such complaints about the story's lack of narrative logic are often dismissed by its fans with Viewers Are Morons, with fans and non-fans talking past each other and on entirely different levels. Literary allegory has been a respected genre since about as long as myths have been written, and it got its start in myths and folklore meant to both explain and symbolize the forces of nature and humanity's place in the world. On the other hand, one can argue that a modern-day author whose story doesn't hold together on every level has only written half a story; Shakespeare's plays may be full of symbolism, but they also work just as well as straightforward comedies and tragedies.
Such stories are often a source of incurable Mind Screw, since there's really no way to make narrative sense out of them, and an Everyone Is Jesus in Purgatory outlook is just about required. When most people think of True Art books, movies and so on, this is usually the sort of thing they're thinking of, though it can all fall down if it's mixed up with too much gratuitous Faux Symbolism. If the fandom insists on coming up with a literal explanation for what's happening, Through the Eyes of Madness or All Just a Dream are common rationales.
See also Rule of Symbolism, which is what happens when an otherwise realistic story relies on a momentary suspension of disbelief in the name of symbolism, and Rule of Cool, which does the same thing for the sake of awesomeness.
This is often an Ending Trope, so beware the spoilers!
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Anime & Manga
Revolutionary Girl Utena runs on this sort of logic, relying on Jungian archetypes alone to explain immortal power struggles, surreal landscapes and a bewildering tournament set in an ostensibly modern, ordinary world. If you've ever found yourself lying awake at night wondering how the heck human girls can inexplicably lay eggs or turn into cars, this might not be the anime for you.
Serial Experiments Lain has the premise that the real world and the internet are gradually merging. It sounds simple enough, but the way it's depicted is entirely this trope. Anything explained is done as a Genius Bonus, but the vast majority of the plot relies on vague symbolism and inferences.
Junji Ito's horror manga usually rely on this sort of explanation for The Reveal. For one low-key example, The Earthbound has people mysteriously losing their will to live and becoming inexplicably rooted to one spot. The revelation that they're being weighed down by the guilt of their secret crimes might not make any literal sense, but it's the only one offered.
Axis Powers Hetalia involves anthropomorphic embodiments of nations living lives that reflect major events in each one's history, initially focusing on World War 2 but eventually moving both forward and backward through history. There are some vague attempts every now and then to flesh out the setting into a more literal, realistic context, but ironically, they only make things more confusing.
Haibane Renmei is centered around the mystery of the setting, and ultimately almost none of it is explained. The details of why don't matter, though, because the important part is the extended metaphor for the important questions of life.
The Sandman operates more on a metaphorical level then on a literal one, to the extent that the physical reality can be said to be an extension of metaphors. Rather appropriate for a series centering around the Anthropomorphic Personification of dreams.
David Lynch's movies have a reputation for falling into this category. Though some of them do have a comprehensible story, there's simply no way to take movies like Eraserhead and Inland Empire on anything except a very symbolic, fever-dream level.
The Shinya Tsukamoto cult film Tetsuo: The Iron Man starts off in a realistic world and gradually delves deeper into Body Horror territory. Like Junji Ito's stories, however, The Reveal for what's happening to the protagonist only makes sense only on a symbolic level.
Lars von Trier's Antichrist also works on this level. Critical discussions on it range the topics of love and hate, order versus chaos and the nature of guilt. Questions about precisely where the talking fox and the tree with hands came from are a different matter.
Identity runs on a symbolic level. The entire world and all the characters are actually nothing more or less than the pieces of a madman's fractured mind, and the struggle between the characters are his internal mental struggle.
The ending of Snow In August, where protagonist Michael Devlin is able to summon the Golem of Jewish folklore and has it beat up the Falcons (the teenaged thugs that have been terrorizing him). He then heals Rabbi Hirsch of his extremely life-threatening injuries (inflicted by the Falcons), after which Hirsch reunites with his wife who had been killed in the Holocaust. The whole thing seems to have been an allegory for the healing power of religious faith and may have also been symbolic of Michael's extreme Character Development, though some have interpreted it as] an Ass Pull.
The Pilgrim's Progress, though it narrowly averts the trope by presenting itself as a dream, remains one of the most famous "pure" allegories in literature. Just in case there's any confusion over the matter, the characters are actually named things like Christian, Good Will, Hypocrisy and Ignorance.
Bunyan also wrote a less well-known allegorical novel called The Holy War, in which King Shaddai and his son Emmanuel fight against Diabolus for rulership of the town of Mansoul. This one dispenses with the All Just a Dream disclaimer, and is considered much less successful.
C. S. Lewis's Pilgrim's Regress, being a sort of personalized version of Pilgrim's Progress, has this.
George Orwell's allegorical novel Animal Farm also works on this level. What kind of farm has talking animals that take over its operations, and what sort of world would keep doing business with that farm as though nothing happened? A farm that represents the Russian Revolution. How, at the end, did the pigs turn into people? Because it represents how a revolution's leaders can eventually become no different than the tyrants they overthrew.
The works of Franz Kafka are also famous for not making sense on anything but the metaphorical level. In his case, though, even the metaphor's very much up for debate. In The Metamorphosis, Gregor Samsa turns into a giant bug. Why that happened and what it represents, if anything, is entirely up to the reader. The most common interpretation seems to be that the transformation represents mental illness and that Gregor is just imagining his transformation due to insanity but it's still debated.
Modernist literature in general approaches its stories from this perspective, most famously in T.S. Eliot's long poem The Waste Land. The images are bizarre, disturbing and can only begin to make sense if the reader takes every everything symbolically.
The Alchemist by Paulo Coelho, about a shephard boy's journey to Egypt, works on this level.
Norton Juster's The Phantom Tollbooth has the tollbooth leading to a world of pure allegory and language come to life (often in puntastic form). There might be magic involved with the tollbooth itself (it's not made clear), but once Milo crosses it, he's off to places like the island of Conclusions (that you arrive at only by jumping) on a quest to return the princesses Sweet Rhyme and Pure Reason to the kingdom of Wisdom. Yeah.
The Prisoner similarly more or less defies a literal explanation, as one early episode after another seems intentionally designed to blow away any rational theory about The Village's nature. The Gainax Ending certainly didn't help matters; it's notable that the 2009 remake resolved the setting by placing it within a literal world of symbols... i.e. a dream world.
Many of the episodes of The Twilight Zone occur on a purely symbolic level as well, mixed in with more straightforward science fiction and fantasy stories. For instance, in "I am the Night, Color Me Black", why hasn't the Sun risen and why is darkness spreading all over the world? Because all the hatred in the world has blotted out the light.
Twin Peaks, being created by David Lynch, also approaches pure allegory at times. There might be supernatural forces at work in the town, or the whole thing might be a prolonged allegory about something else entirely.
Mage: The Ascension has a few elements of this in various places, as well as in its general mechanics. Notably, there is the Astral Umbra, which Mages can access with the sphere of Mind. In this realm, literally everything is the symbolic made manifest, with dream logic, mythopoetic structure, and metaphor acting as the rules of the realm.
Mage: The Awakening is similar in many respects. Not only in its Astral Space, but many aspects of the "real" world are implied to be purely symbolic abstracts made manifest (most strongly in the sourcebook "Keys to the Supernal Tarot"), and Imperial Mysteries depicts the Supernal World as utterly requiring the imposition of symbolism on it to make interaction that does not utterly obliterate human existence to be possible.
Nobilis is built around this trope. In the game, players enact Nobles, figures of power that stems directly from the aspect of reality that they embody. They wield these powers through symbols and thoughts that act on a world that is completely a World of Symbolism. By manipulating these symbols, they are able to create acts in the physical realm which are truly massive.
The Summoning of Everyman, a morality play by Ben Johnson. God sends Death to summon Everyman, who represents mankind, and Everyman meets characters like Good Deeds and Knowledge. 16th century morality plays weren't known for being subtle.
Imitated in the early 20th century by Everywoman and Experience.
Journey. While the story obviously resembles The Hero's Journey, you may feel free to interpret it whatever you want, as no words are heard at all here, even when you communicate with a companion. The backstory is told through confluences and glyphs. The only voices you hear in the game come from cloth creatures and machines, and still no words.
Metal Gear Solid 4 abandons it entirely in order to provide concrete answers to every question the series raised.
The Path runs wild with this trope, as it's a metaphorical horror take on the Red Riding Hood story. Why are six girls named after different shades of red each visiting their creepy grandmother one by one, walking through the woods alone and confronting a shapeshifting being that leads them to a nightmarishly transformed version of grandma's house? Well, it's because the wolf represents temptation, grandma's house symbolizes death (or adulthood), the path is a metaphor for obedience, the girls embody different stages of adolescence...
Rule of Rose is arguably All Just a Dream, but it's still got symbolism coming out its ears, and it's never made clear what most of it means and what actually happened to Jennifer.
The Silent Hill series is famous for working, to some degree, on this level, though how much of it can also be explained via supernatural phenomena varies from game to game theory to theory. The second game in particular presents its story almost solely on a symbolic level. What is Pyramid Head? A reflection of James' guilt, sexuality and need for punishment. Who is Maria? An idealized yet distorted memory of his dead wife. Yes, but, literally, what are they and where did they come from? At the very least, you'll need information from the other games to even start making guesses.
For A Change goes out of its way to avoid describing much of anything with concrete detail. Symbolic interpretations are therefore easier to make than literal ones.
Parodied in Oglaf. A man walks the Path of Vengeance, which includes tasks such a physically grappling with Hope and being handed a small basket full of Justice, until he meets the Allegory for Allegory itself.