Gunter

Gunter is the name of the penguin that most commonly accompanies the Ice King, although in "Princess Monster Wife," it is shown that multiple penguins have subtle variants of that name but the Ice King pronounces the other penguins' names differently. The Ice King uses Gunter as his personal servant, whereas the other penguins working for the Ice King are more or less slaves.

Episode appearances
In "What is Life?," one of Gunter's known duties is being used as a stand for the Ice King's video game machine. Gunter also tried to warn the Ice King that Finn and Neptr were in the room.

Gunter is mentioned in "What Have You Done?" when Ice King says, "Gunter, dispose of Finn's carcass."

Gunter appears again in "It Came from the Nightosphere," where his/her soul is almost stolen and is dubbed "by far, the most evil thing I've encountered" by Hunson Abadeer, and thrown at the Ice King. The Ice King's response is "Gunter, who told you you could fly?"

Gunter appears in "The Chamber of Frozen Blades" and is very sick, but it later turns out that she was pregnant and the egg hatches revealing a floating pink kitten.

Gunter re-appears in the episode "Still" trying to mess with the potions Ice King brought to freeze Finn and Jake. Ice King says that Gunter likes to break bottles, which Gunter eventually does. She is shown to be a troublemaker in this episode.

Gunter's name is written on Marceline's journal in "Marceline's Closet."

Gunter appears multiple times in "Holly Jolly Secrets Part I" and "Holly Jolly Secrets Part II." Notably, during his/her "Penguin Dance Party" video, Ice King calls two different penguins "Gunter" in the same scene. It is unknown whether this was intentional by the show's staff, but this is the main in-show evidence suggesting that Gunter is not a specific penguin.

In "Five Short Graybles," Ice King makes Gunter float out to sea on an iceberg because he believes that Gunter is making his cave smell bad. It also shows that Gunter can play the keyboard, though not very well.

In "Princess Monster Wife," only one penguin is called Gunter and many other penguins are addressed with similar, though different, names.

In "Reign of Gunters," Gunter proves s/he is the most (potentially) evil entity in Ooo, excluding the Lich, by using Ice King's demonic wishing eye to conquer Ooo and destroy all of its glass objects. At one point, the army of Gunters ran out of bottles to break, so Princess Bubblegum suggested reconstructing the bottles to perpetually keep Gunter occupied. Gunter got impatient, however, and attempted to shatter one of the Gumball Guardians' heads (seeing as it is essentially a giant bottle), showing that Gunter is so obsessed with shattering glass that s/he does not particularly care whether they are sentient or not. Gunter is only stopped when the Ice King returned from his outing, scolding her for taking his wishing eye and telling her to go back to the castle.

In "Jake the Dog," Ice King accuses Gunter of taking his crown jewels.

In "Five More Short Graybles," Gunter is among the penguins who want to lick Ice King's salty tears, but Ice King pushes him away, telling him he's on a "low-salt diet."

In "All the Little People," there is a Little Gunter among the Little People. Strangely, he notices Finn's hand.

In "Simon and Marcy", Simon refers to Marceline as "Gunter" when he puts the crown and starts acting crazy.

In "Princess Potluck", Gunter helps Ice King by baking lasagnas to feed the rabbits to somehow ruin Princess Bubblegum's potluck. Afterwards, Ice King dresses Gunter as a princess (using old socks) and tells him to go there and throw fruit punch at everybody, although everybody notices it's just Gunter and start cheering him/her, which starts to party along.

Appearance
Gunter looks like a regular penguin with big black eyes. She/he has a black back, black flipper-like wings, and a white flat belly. She/he also has a pointed yellow beak and yellow feet with three toes.

Sex and gender
There are a number of penguins that Ice King refers to as Gunter, or a variant thereof. He pronounces it differently depending on the penguin's personality. This is shown in "Princess Monster Wife" when multiple penguins are addressed with some variant of Gunter (e.g. Gundy, Gunder, Gunthy, Goonter, Günder...). Also in "Princess Potluck", the Ice King addresses two penguins as Gunther (one he sends to the party and one who helps him with his own), further showing that the name is applied to all penguins in the kingdom. Because of this, it is difficult to ascertain the sex and gender of Gunter, if there is a single Gunter.

It is not completely clear whether Gunter is male or female. However, in "The Chamber of Frozen Blades" Gunter lays an egg, which would likely mean it is female.

Ice King typically refers to Gunter by masculine pronouns. For example, in "The Party's Over, Isla de Señorita," Ice King tells Gunter he is now the man of the house.

Ice King
The Ice King sometimes gets upset at Gunter, such as when he/she accidentally broke his video game or ate his socks, but the Ice King seems to appreciate his/her company. Ice King told Gunter that he loves him/her in "Holly Jolly Secrets Part I." The Ice King often speaks to him/her affectionately like a father. For example, in "What is Life?" Ice King calls Gunter "daddy's little fatty," and he often addresses him/her with terms of endearment such as "honey" or "sweetie."

Kitten
As Gunter's child, Gunter gives a lot of parental affection to Kitten. Throughout "The Chamber of Frozen Blades," Gunter is seen kissing and hugging her egg. Gunter does this too in "Reign of Gunters."

Trivia

 * In the episode "Holly Jolly Secrets Part II," BMO deciphers a complicated code in the Ice King's crying, which reveals an image of Gunter when printed.
 * Gunter (variations: Günther, Gunthar) is an old German name made of the words Gund ("Fight," "Battle," "War") and Heri ("Crowd," "Folk") referencing the fact that Ice King calls all his penguins some variant of Gunter, making them his army.
 * Marceline's journal has the name Gunter on it in the episode "Marceline's Closet."
 * Gunter, according to the Ice King, likes to smash bottles and eat socks.
 * Ice King has a picture of Gunter in his diary in "The Chamber of Frozen Blades."
 * In the game FusionFall, Gunter is a supporting character in the game along with Finn, Princess Bubblegum, Jake, and the Ice King. S/he also has an evil fusion counterpart called Fusion Gunter.
 * Gunter's only known weaknesses are Ice King and the squirty-squirt.
 * Gunter's baby has a special fire shooting power.
 * In "Reign of Gunters," Finn, Jake, and the Ice King continuously refer to Gunter as "Gunther," with an "h."
 * In "All the Little People," Little Gunter is seen following Finn's hand when he puts down Ice King's drums, even though Little People exist in another dimension and they can't see ours. This implies Little Gunter can see our dimension.
 * In "Simon and Marcy," when wearing the crown, Simon refers to Marceline as "Gunter."
 * In "Princess Potluck," Gunter is shown to own, or at least temporarily have possession of, a taser which she/he uses against both the Ice King and Finn.