Cthulhu is a creature first introduced in the horror story "The Call of Cthulhu" by of H.P. Lovecraft, in 1926. Lovecraft added Cthulhu as an omnipresent being, making additional appearances in later stories. He is the greatest and most powerful of the Great Old Ones, a race of godlike beings trapped within the Earth, enemy of the Elder Gods. He is described as being gigantic in size, has squid-like tentacles from its head, claws and wings, though can change shape at a whim. Cthulhu dwells in the sunken city of R'lyeh, dead but dreaming, awaiting its return.
Cthulhu is worshiped and feared by followers and cultists all across the globe. They wait for and dread the day that he will one day Cthulhu will awaken and destroy the world. There is no way of knowing Cthulhu's wants or motivations, as we are merely ants to him. But know that one day he will return when the stars are right, and we will all suffer the consequences.
There are images of Cthulhu found throughout history and across civilizations. Interpretations vary. Even the name of Cthulhu varies, though as Lovecraft has stated "this is merely the closest that the human vocal apparatus can come to reproducing the syllables of an alien language."
It appears to be a species rather than a single entity, as there are many of them to be found throughout the series, starting with Symphony of the Night.
There is a well known error in which the creatures Cthulhu and Malachi had their names switched during localization, perhaps due to their similar sprites. It's clear that both monsters fight in the same way, have similar hovering movements, and sport long, powerful arms. However, Cthulhu is supposed to be the green monster with the octopus face, and Malachi should be the gargoyal-like beast. However, it's the other way around and despite being simple knowledge, has not been fixed despite Cthulhu (as Malachi) appearing in several more Castlevania titles. This wiki covers what is supposed to be Cthulhu, as it appears in the original Japanese release, even though it is called Malachi from the translation on.
Assuming he's named correctly, Cthulhu fights by slowly hovering around an area and dropping down to take large, but slow swipes at foes. It also can summon a projectile which is an amalgum of terrible energy launched from a multi-pointed star, presumably some sort of channeling portal. This attack can be obtained by Soma Cruz and Charlette in Castlevanias Dawn of Sorrow and Portrait of Ruin, respectively. It is called Nightmare.
Sympohny of the Night:
Level: 39. HP: 450. Exp: 666 (lawl)
Weak versus Holy, absorbs Dark.
Drops: Dark Shield, Dark Armor.
Dawn of Sorrow:
HP: 333. Exp: 356
Weak versus Holy. Strong versus Dark.
Soul Type: Bullet. Soul Drop Rate: 4% Max Soul Level: 3
Drops: High Mind Up (8%)
Note: You can use it's soul to synthesize Osafune into Kunitsuna.
Portrait of Ruin:
HP: 44. Exp: 333. SP: 6
Weak versus Holy. Strong versus Dark.
Drops: Skull Mask, Nightmare spell
Cthulhu appears as a creature that can be summoned by name in Scribblenauts. He is a powerful creature that can swim and is likely to attack any other living creature.
In Pray for Death
He was inspired by the H.P. Lovecraft character, though in this game he seems a lot less powerful and smaller.
Cthulhu Saves the World
In this deliberately retro JRPG, Cthulhu is awakes with the intent to destroy the world, only to lose all of his powers when a powerful wizard casts a spell on him. Then (as he learns when he overhears the game's narrator) he must become a true hero in order to regain his powers and destroy the world. Cthulhu is the main protagonist of the game, and in his quest he meets up with other strange folk (including a talk sword and an alien cat who went to college with Cthulhu) who join and fight alongside him in turn based battles. Cthulhu's signature ability to turn people insane is used as an attack in the game that drives enemies insane, which opens them up to more damage, and changes their sprites into kookier versions.
As one might have guessed from this description, the game is very comical and lighthearted in tone, and features many jokes about and references to JRPGs and the works of H.P. Lovecraft.