Cthulhu is the most famous figure of the Cthulhu Mythos spawned by horror writer H.P. Lovecraft. He made his first and only appearance in the 1928 short story "Call of Cthulhu," the definitive cosmic horror story. He isn't that important within Lovecraft's universe, a relatively minor member of its complex pantheon - the real head honcho of the cosmos is the Blind Idiot God, Azathoth, who writhes and gibbers in the nuclear chaos of creation, and Cthulhu himself is pretty much "just" a very powerful alien sorcerer-priest and king of his own species - yet Cthulhu's alien but semi-pronounceable name, the popularity and quality of "Call of Cthulhu," his frequent references in Lovecraft and his friends and follower's stories, and his iconic winged squid-man appearance have cemented his role as the poster-child of the eponymous mythos. He has since gone on to star in the roleplaying game of the same name, as well as another based on the Mythos known as Trail of Cthulhu.
He is very strong against Investigators, and can eat 1d3 of them per round; however, he is weak against boats, especially when they make called shots to the face (and are manned by Norwegians), though his fast healing usually makes up for it.
Fluffwise Cthulhu may be something akin to a C'tan before they got shattered with an Alpha-Plus psyker level, capable to win against a Greater Daemon and perhaps giving an Incarnate a run for his/her money, on the other hand, while he and his progeny managed to conquer a large portion of the Earth they couldn't wipe out the technologically advanced Elder Things neither the Great Race of Yith, with both of them fighting back to a stalemate, followed by peace agreements, yes, you read it right, Cthulhu went to the negotiation table, although he probably was the one giving most of the terms, it speaks a lot of the Yithians and the Elder Things (or very little of humanity, probably the later) that they convinced him to parley.
To be fair the Yithians could time-travel and had tesla weaponry as well as tectonic manipulation devices, while the Elder Things could use molecular perturbation weaponry and bioengineering, and both races knew to use sorcery without having their minds blown out. Wow, this would have made a truly awesome tabletop game or a RTS videogame.
Cthulhu's main advantages are his sorcerous and madness inducing powers and the tyranid-like numbers of his progeny as well as their regenerative powers, all in all he would have no real problems defeating mankind at our current technological and evolutionary level.
Cthulhu's earhtly domain is based on the city of R'lyeh, a massive non-euclidean megalopolis which is currently sunk in the Pacific ocean, the aquatic body effectively blocks out Cthulhu's psychic powers, in order to bring him back R'lyeh must rise, the stars must be right and someone outside must break the seal of Cthulhu's chamber.