Physical Scars, Psychological Scars

"Do you want to know how I got these scars?"

—The Joker, The Dark Knight

When something causes both physical wounds and mental wounds, the former can double as an extended metaphor for the latter.

Not to be confused with Good Scars, Evil Scars, which is about the contrast between how scars are portrayed on good characters and on evil characters.

Spoiler-heavy by its very nature.

Examples of Physical Scars, Psychological Scars include:

Anime and Manga

In Naruto, Ibiki has scars on his head that represent the mental torture he underwent years back.
Kazundo Gouda in Ghost in the Shell: Stand Alone Complex: Second Gig was badly scarred in an accident. He could have elected for reconstructive surgery, but chose to retain his mutilated appearance, probably so he could see the looks on people's faces when they saw him for the first time. Mentally he changed from a bland bureaucrat to a manipulative bastard.
In Gundam Seed, Izak gets a scar while fighting against Kira. He could get it removed, but decides against it. He wants to keep it until he gets his revenge against Kira. In Gundam Seed Destiny he doesn't have it anymore, after giving up on his grudge.
Yu Yu Hakusho: Sensui is covered in scars to point out how utterly insane he is.
Rurouni Kenshin is identified by his cross-scar, and the backstory behind it is what forms the kind of person he is today.
In Berserk Guts is physically maimed during the Eclipse when he attempts to save his lover Casca from being raped by the demon lord Femto by first chiseling off part of his left arm but ultimately fails and is blinded in his right eye by a demon's claw when he tries to push himself up to try and get to her once more. These two scars serve as a painful reminder to Guts of what he failed to save on that fateful day. In fact, he uses the last sight of his right eye as a reminder of how much he wants revenge on the person who took the person who meant the world to him: Griffith.

Film

In Star Wars: Revenge of the Sith, Anakin's relationship with Padme went downhill fast when he said they could rule together; when Obi-Wan Kenobi showed up, Anakin blamed him for her turning against him. They got into a fierce lightsaber battle with Obi-Wan, with Anakin falling into a river of lava in the process, and screaming at Obi-Wan while on fire. He survives long enough for Palpatine's medical team to show up and save his life, and in the process putting him in a full suit with artificial limbs. Basically, his new suit buried his physical scars, while his new role in the empire buried his mental scars.
In The Dark Knight, The Joker subverts this since his different stories about how he got his physical "scars" (which are apparently supposed to be a metaphor for his mental scars) contradict each other, implying in turn that he's probably not being truthful about his mental scars either. Harvey Dent, however, plays this straight because his face was burned in the same incident that killed his girlfriend, Rachel Dawes.
In The Lion King 2, Kovu gets a scar across his eye from his abusive mother Zira, while she blames him for Nuka's death.
In The Princess Bride, Inigo Montoya, has a scar on each cheek given to him by the man who killed his father which serves to strengthen his drive for revenge.
In the 2002 film version of The Count of Monte Cristo, the scars on Edmund's back that he received from being repeatedly lashed in the Chateau D'If represent how his time in prison embittered him.

Literature

In The Jungle Book, Hathi the elephant has a large white scar from the time he fell into a spiked pit trap and felt humiliated enough, so that when he escaped he razed three villages.
The Seablite gang from Dark Life have both physical and psychological scars from their time as Doc's unwilling experimental subjects.
In Cagebird, Yuri's cutting scars represent his change from willing participant to unwilling victim of Falcone's pirates. The very act of cutting itself is used to symbolize times when Yuri's too stressed to even address his emotions in the narration.
A Dance With Dragons: Theon is tortured physically, including everything from probable castration to flaying. The psychological effects include a complete loss of identity, possible insanity, and a rather twisted relationship with his torturer.

Live Action TV

Babylon 5: The character Colonel Ari Ben-Zayn from "Eyes". He has a massive disfiguring scar down the side of his face, he has it in for Sinclair and anyone on his staff, and he was traumatized by war back in the day.
Sweets from Bones has scars on his back from his childhood abuse, which has also obviously scarred him internally as pointed out to Booth and Brennan by Gordon Gordon Wyatt.
- Also Booth himself, who has scars on the bones of his feet from torture in Iraq, and it's a representation of the emotional trauma he suffered during the experience.
Person of Interest: Mr. Finch's limp and fused neck vertebrae, suffered in an unspecified (so far) incident which it is implied, also caused the death of his partner Nathan.

Professional Wrestling

Kane was supposed to have this. His body was to be horribly burned by a fire either he or his brother started, which killed their parents, but the makeup didn't work out when he unmasked, so now they're just emotional scars.

Video Games

Hanako Ikezawa in Katawa Shoujo has severe burn scars from a childhood tragedy and crippling shyness as a result of them.
Fenris in Dragon Age II has scars all over his body from the lyrium used to give him his phasing powers. They also represent his mental scars from years of slavery and abuse at the hands of Danarius, the Tevinter magister that Fenris managed to escape.
In a way, the massively scarred body of The Nameless One from Planescape: Torment functions like this: He is so scarred that he has practically no intact skin left that could tell him how he used to look like, and also has amnesia. Furthermore, he loses his mind every time he dies; the scars indicate that his mind is probably just as 'scarred' from being repeatedly wiped as his body is.

Western Animation

Avatar: The Last Airbender: Prince Zuko's father issues are represented physically through the scar over his eye, and Zuko suffered both as part of a confrontation with his father. The connection between the two is even discussed in-universe, when Katara laments that she didn't get the chance to heal the scar because it may have helped clear up his emotional issues as well.