Sometimes it's hard to live with the thought you could kill, maim or destroy with very little effort, often the only difference between being a mass murderer and terrorist and a hero is being scared of what you are capable of and keeping it in check.
For whatever reason a character (usually a hero or Anti-Hero) is worried about doing more harm than good whenever they use their powers or abilities. Sometimes this only causes a healthy amount caution and stricter set of self imposed moral rules, occasionally it can get to the point of being Willfully Weak and using Power Limiters but sometimes it's so bad that they are petrified with the fear of what could go wrong to the point of a Refusal of the Call, 10-Minute Retirement or becoming a Retired Badass. Occasionally this is the result of not knowing What You Are in the Dark where the little ideas in the back of the mind tempt them to abuse their power or position for the better good at the cost of general well-being or freedoms.
The reason for this fear is what keeps it an interesting bit of character struggle note it's usually moral or sanity based, being between what they are capable of and what they can see properly allowable of themselves, as does how it may or may not be resolved or not. Expect a "World of Cardboard" Speech at one point or another.
Some examples of the reasons for this fear could be one or more of these:
It seems many of the officers of the Gotei 13 feel this way and are taught to think as such.
It's implied to be part of the meaning behind Squad 3's theme of "Despair" as given by Izuru Kira
Izuru Kira: "Battle is not a stage for empty heroics and nor is it something to take pleasure in. Battle is filled with despair. Dark, terrifying. That is the way that it should be. That way, people learn to fear battle and to choose the path of non-violence whenever possible."
Meliodas from Nanatsu No Taizai has a problem with using his full powers and actual weapons on the grounds that he is too powerful for anyone to handle and has an aversion to killing. Case-in-point and what he considers his greatest failure, he completely destroyed the Danafor kingdom in the past when he let his wrath get the better of him.
The characters who refuse to use a philosopher's stone do this in a bit of an almost played with in a way in that the power itself they fear and refuse as is gotten by using up human souls, it's why Dr. Marcoh runs from the military, he is terrified of what they do with his research.
Most of the veterans of Isval are hinted to still be getting over this after what they found themselves capable of in the war, most just hide it really well or convince themselves that they are now using these abilities for the greater good especially Lt. Hawkeye, Colonel Mustang and Major Armstrong (who actually freaked out on the battlefield and broke down crying after seeing what he was capable of.) Also a bit of why everyone is so creeped out by Kimblee, despite it all he completely lacks this fear.
Dr. Knox is a bit of this and why he no longer considers himself a Doctor and only worthy to work on corpses, in his mind Doctors make people better he has tortured with his knowledge.
Why Berthold is so careful about who he shares the secret of flame alchemy with, it's so powerful it can shape the world. in the same vein why Riza insists that Roy burn the secrets to Flame alchemy off her back
The titular Rurouni Kenshin, he only uses a reverse blade sword because he is so damn deadly with a real blade and has already caused so much damage, also why he is determined to be the last of the Hiten Mitsurugi-ryu kenjutusu
In the manga version of Battle Royale, Hiroki Sugimura has a case of this. He is a very powerful martial artist who feared that he would accidentally hurt people, or worse yet would come to enjoy hurting people in fighting, he went the route of being a Technical Pacifist instead and in the past would even allow himself to be bullied rather than stand up for himself. He would only get involved in fighting if somebody else was in danger of being hurt.
When Lina Inverse of The Slayers learns what her secret spell, Giga Slave, really is about (it is about requesting intervention of an unpredictable, chaotic creator overdeity who may destroy the world on a whim), she goes through a period of being afraid of her own knowledge. This is, however, temporary and in the fifth season, she is once again ready to cast Giga Slave.
Superman, Depending on the Writer his varying strength is attributed to mental blocks he has consciously put up to keep his power manageable. Also one of the trope codifiers as often it is shown that what he considers one of the biggest threats to the world is himself going rogue or being put under some kind of mind control, which is also why he considers it a good thing his best friend Batman keeps kryptonite in his arsenal at all times.
In the Post-Crisis era comics, Batman's first chunk of Kryptonite was actually given to him by Superman for exactly this reason.
X-Men. Cyclops, Depending on the Writer , which is why Professor X says he put Scott in charge of the first group: he applies the same caution and restraint to the other's abilities that he does to his own.
Bruce Banner has this attitude about the Hulk, also a possible interpretation of Hulk's quote from 'World War Hulk'
Hulk:"I'll hate you forever. Almost as much as I hate myself."
Spider-Man. Spidey has some shades of this. Notably he doesn't drink because he's afraid of loosing control of his super strength and after a certain incident involving Gwen Stacy is crazy cautious about calculating the force necessary to safely catch someone with a web line. He also occasionally seems to believe this is part of With Great Power Comes Great Responsibility.
After Doomsday develops sentience he gets hints of this, though part of his power is "evolving out" his weaknesses every time he dies and revives it could also be a natural fear of death, though he does get better with it to the point that in an alternate timeline he fits this to the point where he continues Superman's legacy in fighting an alien army. (his sentience was wiped out when the alternate timeline was unwritten).
Lennie from Of Mice and Men is sort of this at times, when ever he accidentally kills a small animal he is shown being very remorseful and promises to be gentler next time (which is an after the fact example), and he really does try to be gentle, he also forgets how strong he is when not actively thinking of it, but how he waited to be told to fight back as he was being attacked by Curly could have been this.
In the Inheritance Cycle Eragon is told that the dangers of learning one's true name is two fold, either you Go Mad from the Revelation when knowing yourself so completely good and bad or you become crippled by fear with the idea of having such a perfect and complete control over yourself (and if the True name is not yours having over another).
The "block" that wilders develop, preventing them from using their power except under particular limiting circumstances, arises from a similar fear.
In The Earth's Children series Jondalar is afraid of the strength of his emotions, so he has trained himself to not get too attached to anyone. (In the backstory he had severely injured another man when the other man had caught Jondalar getting emotionally attached to a woman who it was culturally taboo for him to connect with.) This changes once he meets Ayla
Part of the Hand Wave for why Mega Man (Classic) and Mega Man X lose almost all of their upgrades in between consecutive games; they have the potential to be unstoppable in power, and they know it all too well. The latter is also a pacifist by his upbringing.
In The Order of the Stick after Vaarsuvius realizes how far reaching the Familicide spell was he/she has a Heroic BSOD made of My God, What Have I Done? and this, saying something along the lines of "This is the ultimate power I spent most of my life looking for? How many other innocents did I kill whose only sin was loving a stranger?"
During the episode "The Avatar State", Aang is haunted by how scary and destructive he can be when he's in Avatar State; being that he's a pacifist by nature and upbringing, it's very justified. In the episode he even says "I was scary" referring to the avatar state.
After Aang burns Katara when he gets impatient with his firebending lessons, he swears that he will never firebend again, this comes up two more times, once in his training with Guru Pathik (who comments that it was the thing Aang let go of the hardest) and when he learns from the masters of the Sun Warriors, only letting it go when the Chief tells him Fire is not only destruction but life too.
Aang's stopping of himself and his past lives from killing Ozai could be considered this or sticking up to his beliefs of pacifism and no killing but the fact that the horrified look on Ozai's face is the thing that seems to trigger it, it could be construed as a factor.
Also Katara's reluctance to use blood bending is an example, she sees it as morally reprehensible and feeling wrong "Reaching inside someone like that."
How Terra from Teen Titans felt about her powers when she first appeared which caused her to freak out and cause even more damage; "Don't lose control, Don't lose control" it's what eventually drove her to Slade where she got over both her control issues and her fear.