My Hero Academia

Mutant
In universe the name for a super power is called a Quirk. Quirks are generally unique to their user, and are classified in multiple categories.

The first person to manifest a Quirk was a newborn baby in the city of Quig Quig, China, who had the ability to emanate light from their body. After that incident, many people around the world began to manifest different kinds of special abilities, though the cause of the phenomenon was unknown. Currently, 80% of the world population possesses a Quirk.

Emitter
Emitter-type Quirks have the ability to either release certain substances, or alter materials around them in certain ways. Emitter-type Quirks usually require a conscious effort to activate and some even require large amounts of concentration in order for them to remain active.

Emitter-type Quirks can also have various ranges. While some users require physical contact to use them others can have much longer ranges with no distance restrictions. While most Emitter-type Quirks users generally have a handle on their own "emissions", it's also possible for users to hurt themselves from coming into too much contact with their quirk. The emissions of Emitter-type Quirks can have a wide variety of properties and abilities.

While some substances produced by Emitter-type Quirks have common properties, some are unique to the user, and thus have special attributes to them. Those who alter the properties of materials around them can also widely vary in effects, from molecular alteration to molecular destruction. Emitter-type Quirks are, by a wide margin, the most common type of Quirk.

Transformation
Transformation-type Quirks cause the user to take on a temporary alteration of some kind. Transformation-type Quirks allow the user to temporarily "transform" their body in a variety of manners, sometimes enhancing existing features or perhaps adding new features to the body altogether. Transformation-type Quirks generally affect the user exclusively, the only known exception being.

Transformation-type Quirks usually require a close-range proximity in order to fully utilize them in combat due to them only affecting the user's body. Transformation-type Quirks are also the only known type of Quirk that can bestow the user attributes they do not normally exhibit by default. Transformation-type Quirks usually require a conscious effort by the user to "activate", though users can be trained to initiate them reflexively. While Transformation-type Quirks are often quite powerful, their biggest weakness is that their effects are often temporary, and over-exertion of said Quirk usually leads to negative repercussions on the user's behalf.

Because this type of quirk combines the activation time and mutant-shape aspects of the other Quirk types, it is also often known as Hybrid or Composite. Transformation Quirks seem to be the least common type of quirk.

Mutant
Mutant-type Quirks can have a wide variety of features and abilities depending on the individual, but the common similarity seems to be that these Quirks are a part of the user's physicality. As a result, these Quirks are always active, but generally, allow the user the greatest degree of control.

Mutant-type Quirks cause the user to exhibit irregular features that generally have some sort of purpose to them. Whether it be bodily alteration or enhancing preexisting abilities Mutant-type Quirks even have the ability to channel certain aspects of their user through them similarly to Emitter-type Quirks. While not necessarily a weakness, Mutant-type Quirks cannot be turned off, and sometimes cause the user inconvenience due to their malformations.

List of those with Quirks by appearance
M = Manga A = Anime

M:01 | A:01 - Katsuki Bakugou ~ Explosions

and friends

M:01 | A:01 - Luminescent Baby Qingqing City, China ~ Luminencance from since birth.

M:01 | A:01 - Person who is part fire.

M:01 | A:01 - Girl whose hands were encased in ice

M:01 | A:01 - Boy using telekinesis to read a book

M:01 | A:01 -  Purse Robber ~ Monster form Quirk

M:01 | A:01 - Shinji Nishiya / Kamui Woods ~ Lumber: Generation and control wood from his person.

A:01 - Deathgoro ~ Super Strength.

A:01 - Bakku Dorafuto / Backdraft - Hydrokinesis.

M:01 | A:01 - Yuu Takeyama / Mount Lady ~ Gigantification: Growth to 2062 meters