Correct spelling, explanation: famous is an adjective formed from the noun fame. It was created in a very common way: by adding the -ous suffix to the basic word (excluding the last vowel). We can’t change its structure, that’s why famouse is just a misspelling.

Definition of famous:
adjective, much talked about, having fame
My grandfather was famous in our city. Everyone knew who he was.
I don’t get why so many people want to be famous. I personally think it’s a burden.

Incorrect spelling, explanation: this mistake affects not only the spelling but also the pronunciation, that’s why it must be avoided. Remember that adjectives are made by adding a specific suffix to the root. In this case, the ending is -ous (like in the following words: dangerous, generous, marvelous, etc.), not -ouse. Thus, the only correct form is famous, not famouse.