Cthulhu is a creation of the writer H.P. Lovecraft and first appeared in his story The Call of Cthulhu. He later appeared or was mentioned in other stories by Lovecraft and also in stories by other writers.
Cthulhu himself is an alien being of immense power who, along with others of his race, arrived on Earth millions of years ago; long before humans even existed. He and his people lived in the city of R’lyeh which existed on an island in what is now the Pacific. For one reason or another, the island sank to the bottom of the ocean. Cthulhu was trapped in the city when it sank and so now lies, sleeping, at the bottom of the ocean.
Cthulhu can only exist fully when “the stars are right”. The exact meaning of this is unclear but at the moment they are not. Because of this, Cthulhu cannot be active at this time and so can only sleep within his sunken home of R’lyeh and dream.
Because he is so powerful, his dreams can sometimes be picked up by humans. Particularly sensitive people (like artists or madmen) are more sucesptible to picking up these dreams. Some of them interpret these as messages from a “god” and attempt to do Cthulhu’s bidding based on them. Whether Cthulhu plans this or if he is even aware that it is happening is unknown.
The thing you must realize about Cthulhu is that he is truely alien. His motives and motivations are not just unknown to us, they are truely unknowable. We do not and cannot understand what he wants or why.
Were Cthulhu to awaken he may destroy the human race or even the world, but not for any reason we could understand. In fact, he may not even be aware that the human race exists. It would be no different than you walking along the street and stepping on an anthill. You could kill hundreds of ants and destroy their “world” and not even be aware that you had done so. The same is true of Cthulhu and humanity.
As I alluded to earlier, some humans are worshippers of Cthulhu but it is doubtful that he is aware of them or that he would care even if he was. There are other species which worship him (such as the Deep Ones) but it is unknown if he has any more interest in them than he does with humans.
How do you make a story about Cthulhu interesting? Go track down a copy of Call of Cthulhu and read it. (It’s available on the web but I won’t link to it since the status of such copies are posibally questionable. You should be able to find it without much trouble though.)
I’m sure others will be here with more information shortly, but this is my take anyway.