The primary reason for ghosts to be Barred From The Afterlife. They want revenge, their story to be told, or simply to be informed once and for all that they are, in fact, dead. To get rid of the ghosts, the hero or heroine will have to either do extensive research in old newspaper articles or communicate somehow with the ghosts. Ghosts with Unfinished Business almost always have Purpose Driven Immortality; once their business is finished, they vanish in a flash of light.
While the futility of revenge is often An Aesop for the living, it seems to be different for the dead.
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In Ranma ½, the anime-only ghost Kogane, who is cryptic about what she needed Ranma and Akane to find because she didn't remember what it was herself; it eventually comes out that she forgot to collect the plush toy that came as a free gift with the purchase of her school day-planner.
In AIR, Shiraho the miko possesses Kano so she can use her to tell her tragic story to someone who will listen. The unfortunate part is that only about part of her spirit is in Kano, and her attempts to share her tale come out as broken speech and Kano trying to re-enact Shiraho's suicide without realizing what she's doing. Yukito finally uses Kanna's feather to unite Shiraho's ghost, hear her out and dispel her from Kano.
And while we're on the subject of the Jidai Geki and modern age intersecting, Ryuuya and Uraha are continually reincarnated out of hope that one of their future lives will save Kanna from her own reincarnation curse.
Hayate the Combat Butler parodies this, having the Priest's ghost say that the last thing binding him to the world was his lingering desire to flirt with a Meido. After convincing him that he can't flirt with Maria, Isumi volunteers to masquerade as one, going on a short quest to find the essence of being a maid. In the end however, he reveals that he was joking and has no intention of moving on yet.
Mahou Sensei Negima! parodied this along with other ghost-related tropes during Sayo's chapter. After learning that all Sayo wanted was to make friends, Negi and Asakura approach her and tell her that she can be friends with them, and she immediately fades from their sight. While Negi tearfully looks up in the sky assuming that she's finally left in peace, Mana, who has a stronger ghost sense than Negi, points out that she's still right there, on the same spot.
Some believed she would pass on if she actually graduated. In the final chapter, it's revealed that even after graduating she didn't pass on and stayed working as Asakura's partner.
In Shikabane Hime, this is what causes Shikabane to occur. Generally, instead of having their story told, they get their brains exploded.
Hell Teacher Nube - Nube's always assisting spirits to pass on to the next life (and tells a spiritual-power seeking student that being able to see ghosts is a terrible power to have since they'll never leave you alone if they think you can help them).
In Rinne, this is what drives part of the series. The title character spends his time helping finish it.
Two exist in Mahoromatic. In episode 5, one of the male students mentions a girl who died after falling down from the rooftop of the school, and her hair dyed red from her blood. The other was a boy who suffered a fatal heart attack after being tossed into the pool by bullies. Although they thought those were just urban legends, said ghosts actually do exist, and Mahoro runs into both of them. However, being a combat android, she's not afraid of either one and is barely aware of the concept of ghosts, but when she talks to them, they both mention being stuck there due to waiting for someone. So she helps them by bringing the boy to the girl inside the school, where they both ascend to the sky and fade away.
The opening theme of Hikaru no Go actually says that Sai came back from death to achieve the "Hand of God" (a move in Go) being that (and wanting to play go even in death) his unfinished business, he and Hikaru even believe so. It's actually a subversion; the real reason he was allowed to came back was to be Hikaru's mentor.
A Rumiko Takahashi short story called "Reserved Seat" was about an old woman possessing her surly rocker grandson after her death, in order to continue enjoying Takarazuka musicals with her neighbor, a nun. The grandson's friend thinks it's because he blew off her funeral and insulted her at the wake, but the nun states it's because the grandmother has a reserved seat until the end of the season. As it turns out, her actual business was to finally see one more show with her grandson, of his own free will. He used to love those musicals when he was a kid, until he accidentally ripped his grandmother's favorite poster and she went ballistic.
In one Simpsons Treehouse of Horror comic, the Simpsons are killed and return as earthbound spirits. While Lisa guesses that their unfinished task involves forgiving a guilt-stricken Grampa for accidentally killing them (he was fiddling with the reception on the TV and accidentally fried them), they settle for scaring the townspeople out of the house (Homer having willed their home to all their friends).
In The Bird Grip, the fox that helps the hero reveals at the end that he was a dead man whose burial the hero had paid for.
Casper the movie. The idea was actually followed through upon to dispatch the Big Bad of the film; she came back as a ghost to steal Casper's father's "treasure", and succeeded - but doing so resolved her unfinished business, and she was hurled back to the afterlife seconds later.
Even though not technically dead, the Bride from Kill Bill was beaten to a pulp and then shot point blank in the head. Her story could be taken as a living version of this trope
You and I... have unfinished business...
Pretty much the entire plot of the 1986 Charlie Sheen movie The Wraith. A teenager is murdered by a local gang of criminals because their leader had a Villainous Crush on his girlfriend. He comes back from the dead in a different body, and travels around in a Cool Car wearing a black suit with a motorcycle helmet. He kills all of them one by one, gives his car to his younger brother, and finally leaves with his girlfriend.
In the Discworld novel Wyrd Sisters, Verence's ghost cannot rest until his destiny has been resolved (Death tells him this, but can't offer any advice as to what it means). Judging from the number of dead kings hanging around the castle, this rarely happens.
On Discworld zombies are also obsessed with Unfinished Business: Baron Saturday wants to see his daughter claim her title, and Lady Lilith dealt with; Reg Shoe wants to see a new world order of justice for all; and Guild of Lawyers boss Mr Slant wants to get paid for conducting his own defence in the trial that led to his execution.
And in Men At Arms, Lance-Constable Cuddy proclaims that his tortured soul will walk the world in torment until he's properly buried. Lampshaded in that Death points out that this is unnecessary, yet the ghost insists his soul can do this if it wants to.
In the Dresden Files book Ghost Story, Harry chooses to become one of these to uncover the mysterious circumstances surrounding his death.
The reason for both ghosts existing in Dirk Gently's Holistic Detective Agency; the ghost of Gordon Way needs to complete his last answering machine message to his sister, and the alien ghost wants to undo the accident that killed his crew and started the formation of life on Earth.
In the radio adaptation, it's briefly three ghosts, as the mother of Michael Wenton-Weakes needs to let someone know that it was her son who murdered her.
In the Mediator series by Meg Cabot, this is the reason most if not all ghosts remain on earth for the heroine to deal with.
The titular spirit of Galaxy of Fear: Ghost of the Jedi should have rejoined the Force, like most dead Jedi. He had no apprentices or family to advise. But his death let Darth Vader destroy an ancient Jedi library, so Aidan feels he doesn't deserve that and must haunt the space station. The fact that someone set up a fake library that drained the Life Energy of people who touched the books, then set up rumors that the Jedi library was still there in hopes of getting to study Jedi essences, and Aidan couldn't do anything about it or even get noticed didn't help. There's nothing to be done for the library, but he does help Tash, the first Force-Sensitive to arrive, to thwart the man and survive, raising his self-esteem enough that he passes on.
In Army of Terror, the wraiths of Kiva are all that remains of its people after Hoole and Gog's experiment wiped out all life on the planet. They exist only for the opportunity to take revenge against the one they hold responsible Hoole. At the climax, the wraiths see a recording of Gog telling the Emperor that he is going to let the experiment continue knowing what would happen to Kiva for weapons research purposes while keeping it a secret from Hoole. After a few seconds of doubt, having cursed one man for their deaths for so long only to learn he wasn't truly responsible, they immediately attack Gog, demanding that he die for his crimes. By the time it's over, there is no trace of Gog or the wraiths.
In A Song of Ice and Fire, both the characters we've met who have been restored to life by the fires of R'hllor have been single-mindedly obsessed with fulfilling the last goal they had had before they died, to the exclusion of all else; Beric Dondarrion with harrying the Lannister forces in the Riverlands, and Catelyn with getting vengeance against the Freys. It's not clear whether this is a supernatural effect, or if dying and coming back just gives one a stubborn streak.
One particularly odd example was the story in which a ghost wouldn't move on until someone brought her a cherished whistle. Well, if it floats your boat...
In Smallville, the spirit of the seventeenth century witch Isobel leaves Lana's body only after exacting revenge on the descendant of the woman who had her killed. This is an unusual example, however, as in previous episodes Isobel showed absolutely no desire to kill this woman.
And in Tomb, Chloe is possessed by the ghost a girl to find and take revenge on her murderer.
Medium's whole premise is that ghosts are getting in contract with the lead psychic character (or sometimes her developing psychic children and/or her psychic half-brother) to help catch their killer or something else, but it’s normally "Catch the killer please".
On the other hand, Ghost Whisperer isn't much about catching killers as conforting the relatives left behind so the ghosts can stop blaming themselves for... errr... being dead, and can move into the Light.
A variation occurs in a Tales From The Darkside episode called "A Case of the Stubborns". An elderly grandfather (the wonderful Eddie Bracken) passes away — but comes down to breakfast anyway, refusing to believe he's dead. Despite all the evidence his family shows him (he doesn't leave breath on a mirror, for instance), the old coot just won't accept it. He's more of a zombie than a ghost, and his condition is due to his cussedness rather than unfinished business, but he still won't "move on". As Grampa's body starts to decay and he begins to attract flies, someone puts a spoonful of black pepper into his handkerchief. The next time he wipes his face, the pepper makes him sneeze. He finally admits that he has died, and "lays down his burden". He was convinced because the sneeze blew his nose off.
In The Tudors Henry's first three wives visit him. Whether they are real or just Henry's hallucinations is up for debate, but all three claim they've come to see their children and to give Henry a piece of their mind regarding how poorly he treated them in life.
About midway through that season, there is a scene where Charles Brandon interacts with the ghost of a nobleman whom he was in charge of having executed(on Henry's orders) for participating in the Pilgrimage of Grace(the only serious rebellion in Henry's entire reign). With hindsight, it's now pretty obvious that the entire point of that scene was to lay the groundwork for the scenes that the above troper is referring to, and to establish that they are, in fact, real.
In Supernatural most ghosts tend to be fueled by rage over the manner of their death and burning their bones is the only way to stop them from hurting innocents. Ghosts with Unfinished Business tend to be after revenge but once they achieve it they will just find a new justification for sticking around. When Bobby becomes a ghost he is told that ghosts can only affect the material world if they become extremely serene or if they channel massive rage. Most ghosts just stick around and fade away without affecting the living.
Bobby himself originally decided to stick around as a ghost to help Sam and Dean fight the leviathans. However, the brothers soon become worried that he is being overtaken by his desire for revenge against the leviathan who killed him. They are afraid that if they actually manage to defeat the leviathans, he will be too far gone to move on willingly.
When they start worrying about him, he keeps insisting that he can control it. But he realized he almost killed Dean and Sam, and tells him to burn his whisky bottle so he can cross over before hurting anyone else.
In Life On Mars, Sam Tyler returns to the real world as he initially wanted but then realizes that he preferred his world with Annie, so he commits suicide by jumping off the police station.
The ghost-based games of the New and OldWorld of Darkness systems take this trope and run with it:
The wraiths of Wraith The Oblivion have a number of ties that keep them connected to their old life, such as Passions (emotions and drives that defined them in life) and Fetters (emotional connections and objects of importance). Most wraiths seek to keep their Fetters protected so they can maintain their power, but quite a few attempt to "resolve" their Passions and Fetters so that they can Transcend and move on.
Orpheus often has agents of the Orpheus Group attempt to resolve a ghost's Unfinished Business to get rid of them, unless they become too violent or dangerous to deal with. In addition, there are a number of powerful, strong-willed ghosts who act as agents for the company, usually because they too have something they need to accomplish before they can move on.
In the New World of Darkness these traits are split between ghosts and Revenants. Ghosts (which appear to, for the most part, merely be psychic echoes left behind by departing souls, with little to no sentience) possess fetters (renamed "anchors") which represent things of significance to the dead person which their ghost is metaphorically attached to (basically things which would have held part of their minds back while their souls were moving on). While resolving a ghosts issues can be used to set it free (allowing it to move on), it is just as easy to simply destroy the anchor (particularly since some ghosts might not actually have issues which can be resolved). Revenants are corpses reanimated by souls which refuse to move on because they possess a Passion, from which they derive will and power. They also can be gotten rid of by simply destroying their bodies (or helping them resolve their Passion).
Geist: The Sin-Eaters has clarified the role of Anchors for ghosts: as long as a ghost is tethered to the world by Anchors, they're unable to change. They may stagnate and spiral downwards, but they can't grow. Once their Anchors are broken or resolved, they migrate to the Underworld, where they're able to develop once more. As such, Sin-Eaters know rituals that allow a ghost to pass on to the Underworld instantly once its business is resolved, and there's an entire Archetype defined by aiding restless spirits settle matters. Resolving the business of ghosts has a benefit to Sin-Eaters that goes beyond charity, however, as helping a ghost to pass on refills their pool of Plasm.
Van Richten's Guide to Ghosts recommends that Ravenloft monster hunters help non-malevolent ghosts to complete their Unfinished Business in order to end a haunting. For Evil ghosts, a standard ass-kicking is the preferred means of disposal, but discovering what Unfinished Business they'd left behind can provide clues to their vulnerabilities.
This is half the plot of Another Code, as you have to help D regain his memories to get the best ending. In the sequel, this will also show up near the end of Matt's sub-plot as well.
In Final Fantasy X, Auron only hung around after his death because he'd promised Jecht to take care of Tidus.
And to make sure Yuna grew up somewhere peaceful and free of corruption.
Final Fantasy X tends to really go nuts on this trope. Yuna's task is not only to defeat Sin, but to send those who died violently into a peaceful afterlife instead of going crazy and becoming fiends. After Sin is defeated, she still ends up as midwife to the dead for Shuyin and Lenne. Spira itself is crawling with plenty of "Unsent" who get caught up in Unfinished Business and forget they're dead!
Even without being officially a Summoner anymore, Yuna still ends up having to essentially send more than half a dozen named unsent to the after life. Whoever operates as the Grim Reaper in that world must be super lazy...
This trope isn't exactly uncommon in fantasy settings, but its presence in this particular game is somewhat jarring considering we are told near the beginning that the dead, if left unexorcised, are supposed to turn into monsters.
The point of Amber: Journeys Beyond was to help three ghosts finish their Unfinished Business, realize that they're dead, and send them across. The World War 2 widow is implied to have reunited with her husband, and the drowned child seems happy to no longer be lost and confused. The crazy gardener thinks he's boarding the UFO he's waiting for, and it turns out he's going anywhere but.
In Chrono Trigger, there is a side quest in a ruined castle where Cyrus (Glenn/Frog's friend) is one of these. Completing the quest grants Glenn his most powerful sword.
As I lay dying, my heart burned with the thoughts of those I left behind.
In Ace Attorney, Mia hangs around because she wants to council Wright and take care of her little sister Maya. Dahlia Hawthorne goes along with Morgan Fey's plot to kill Maya because she wants to hurt Mia, even though Mia herself has been dead for years.
the white chamber has an interesting case of this. Arthur, who was murdered by Sarah, somehow managed to activate the artifact and live on through it, using its power to force Sarah through a loop that would only be broken by Sarah atoning for her sins. When she does, Arthur blows the station up after letting her escape.
The Celestrians of Dragon Quest IX can see ghosts and help them to move on by taking care of their unfinished business.
One of Runescape's low level quests has you use an amulet of ghostspeak to help a ghost pass on. The amulet is later used to grant an entire town of ghosts the freedom to pass on.
A side quest in Dungeon Siege II teaches you a chant that lets you talk to spirits so that you can solve their problems in further side quests.
In the PC game The Stroke of Midnight, an American author staying in an abandoned Romanian castle finds herself watching the unfolding Love Triangle of the castle's last heir and the two sisters who were vying for him. He and the one who genuinely loved him are trapped in the castle until the Player Character's actions enable them to reunite and move on. The other sister was directly responsible for both of their deaths.
The second Dark Parables game has two ghosts. One, Princess Ivy, is benign. She's the Frog Prince's first wife, who is watching over her immortal husband and trying to help the player character end his curse. The other is Snow White - yes, thatSnow White - whose business is still unfinished. The third game then reveals that Snow is actually still alive, but is now the Snow Queen and carries Irrational Hatred for her ex-husband.
In The Blackwell Series, the reason why some people stick around after their death varies. But in most cases it is because they have Unfinished Business, combined with the fact that their death is usually such a traumatic experience, that they are suppressing the memory of it, often leaving them with a fractured mind and doing the same actions over and over again. More often than not, helping the ghosts to reach a conclusion to their repeated action is the key to make them realize that they are dead, and helping their soul to move on. This is what the Blackwell family does.
The Dark Pieces in the Magical Girl Lyrical Nanoha As Portable games are Dream People created from memories of the past, with their hopes, wishes, and dreams being their anchor to existence. The sequel even revealed that dead people could temporarily live again through this. There are two ways of eliminating them, by defeating them in battle, or more rarely, by allowing their final wish to be fulfilled. Dark Piece Rynith is the latter. Having been dead for three years, her wish was to make sure that her two charges, Fate and Arf, were still fine after she died. After seeing that they're both stronger in mind and body, are now surrounded by many friends, and are even saving the lives of other people, she bids a final farewell and Disappears into Lightdespite the tearful protests of both Fate and Arf.
Rynith: Thank you, Fate, Arf. Meeting you and spending time with you have made me very happy. Using the magic I thought you... Thank you. I wish for your happiness from beyond the sky. My precious Fate and Arf... Goodbye... Thank you.
Tales Of Monkey Island: In the first half of Chapter 5, Mighty Ghost Pirate Guybrush needs to find a way to return to the living world from the Crossroads (he even lampshades this when he talks to Morgan, also a ghost); once he finds a way, he realizes that he can't destroy LeChuck or save Elaine this way in the living world, so he needs to repossess his corpse to finish what he started.
In Echo Night, the various shades Richard encounters have something like this tying them to the Orpheus. He just has to figure out what and complete various tasks for them so that they can move on.
Occasionally, the Ghostbusters didn't actually have to zap the ghosts to bust them. By helping them accomplish their Unfinished Business, the Ghostbusters made the ghosts rest in peace, which worked just as well as blasting them with the proton beams.