Malevolent Mutilation

There are many kinds of Body Horror. Some are done from a Painful Transformation, Gorn, Orifice Invasion, Chest Burster, or something else terrible happening to a person's body.

This is when a person's body is mutilated due to deliberate intent.

Such a character has been subjected to body modification (tattooing, piercing, cutting, etc.) to a degree that goes beyond "fashionable" and straight into Squick. The kind of body modification that would make a person look like a monster, and their onlookers wince in imagined sympathy pain.

Why a character would be subjected to such extreme modifcation is varied. They might have been an innocent who was forced into such transformation, in which case they might be The Grotesque. Unless the pain of their modifications is such that it drives them to lash out. Alternatively, it may be a sign that a character is Obviously Evil, especially if they did this willingly to themselves. Evil Is Visceral after all, and there is little more visceral than mortifying one's own flesh.

If a character has a few obvious piercings or tattoos, they are not part of this trope. This is only when those body modifications are taken to such a torturous extreme only a twisted mind would want to inflict or recieve them.

Anime and Manga

A downplayed example, Kazundo Gouda in Ghost in the Shell: Stand Alone Complex: Second Gig, a villain who has a horribly mangled face and strangely-shaped head due to an accident. Although the damage was not voluntary, it's explicitly pointed out that he could repair his face using the world's futuristic technology and his wealth, but makes the choice not to. He claims it helps make him more memorable.

Film

The Cenobites Hellraiser are an iconic example, being an order of extra-dimentional Sense Freaks who seek to share their "pleasures" with those who can solve the Lament Configuration puzzle box.
Darth Maul in Star Wars: The Phantom Menace, who has his entire face covered in black and red ink to present an imposing appearance.
Karl Ruprecht Kroenen from Hellboy, so very much.

Professor Bruttenholm: Suffered from a masochistic compulsion commonly known as surgical addiction. Both eyelids surgically removed, along with his upper and lower lip. The blood in his veins dried up decades ago. Only dust remains. What horrible will could keep such a creature as this alive?

Milton Dammers the obsessed FBI Agent in The Frighteners opens his shirt at one point to show that his chest is covered in scarification, presumably from his time spent undercover with various cults.

Literature

The New Jedi Order trilogy from the Star Wars Expanded Universe presents the Yuuzhan Vong, an extra-galactic species who revere pain as a way of life. Many of their religious ceremonies involve cutting, piercing, and scarring, that they might better experience the pain they believed their gods suffered to create the universe. Promotion ceremonies involve having their limbs and organs removed and replaced with artififically grown versions.
Just about every vampire in the Necroscope series, especially in the "Vampire World" books. The Vamps there deliberately twist and torture their own flesh to make them as horrific as possible, it is something of competition amongst male vamps to see how can have the worst.
Hemalurgy in the Mistborn books involves pounding metal spikes into your body. The Steel Inquisitors, who used Hemalurgy the most, had railroad-sized spikes driven through everything but their noses, including two so far into their eye sockets that the points come out the back of their heads.
In Chasm City, Orcagna has a large square hole cut through his abdomen.
The Blood Pact opposing Gaunt's Ghosts during the Sabbat Worlds Crusade are so named because they ritually cut their hands against the jagged edges of their patron's Power Armor to seal their loyalty to him and their solidarity as fraternal warriors. They also tend to engage in further scarification, cutting chaotic glyphs into their faces and scalps in order to demonstrate their dedication to the Ruinous Powers. Even those tend to be covered up by the iron masks they wear which are shaped into grotesque and monsterous faces.

Live Action TV

The Reavers from Firefly. The way they scar themselves and jam random bits of metal into their flesh is the least horrible thing about them.
In The X-Files episode "Humbug" they meet The Conundrum (played by The Enigma) at a carny show. He has a jigsaw-puzzle themed tattoo covering his whole body.

Tabletop Games

Members of Clan Tzimisce in Vampire: The Masquerade that make heavy use of their clan Discipline, Vicissitude to modify their bodies are prime examples of these.
This is true of many Phyrexians in Magic: The Gathering. In fact, one such creature, Elesh Norn, is known as the "Grand Cenobite."

Video Games

Soulblighter, The Dragon and Dragon Ascendant from Bungie's Myth: The Fallen Lords and Myth II: Soulblighter, respectively. He walks about stripped to the waist to display a long snaking scar on his torso where his heart was removed in a magical ritual that made him Nigh Invulnerable. More obviously than that, his nose and lips were cut away from his face, leaving him with a permanent rictus grin.
The monsters in Amnesia the Dark Descent. They were all once men, subjected to a Painful Transformation and subsequent modification, such as the removal of their jawbones and hooks and other jagged pieces of metal being shoved through their flesh to use as weapons.
Star Wars: Knights of the Old Republic 2 featured the villain Darth Sion, a Sith Lord who learned to better draw on The Dark Side through his own pain. Medical scans indicate that every bone in his body has been shattered and re-knit together, and his skin is grey and leathery, with many cracks running across it as though calcified and broken. Ultimately, it is the strength in the force he gained by doing this that is all which holds him together anymore.

Web Original

The SCP Foundation has quite a few of these, notably the Cyborg Child and the Tortured Iron Soul.

Real Life

Foot binding. Fullstop. Causing painful, permanent deformity to children against their will in the interest of "beauty" is pretty much the definition of this trope.