People in America ask "how are you" a lot. Many people would reply with "I'm good." If I am feeling OK, I would say "I'm well." Which version is right?
"I'm fine" is the standard happy medium. "I'm well" marks you as British English, while "I'm good" marks you as casual American. "I'm good" is incorrect in the traditional sense, but is broadly used to affect careless informality.
"Good" is traditionally a moral qualifier, or an indicator of ripeness, not human wellness as such.
Both are grammatical in that they are on the pattern Subject – Verb – Complement, one of the seven patterns found in English sentences. Well and good are both adjectives (although well can also function as an adverb). I'm well suggests the speaker is in a good state of health, while I'm good, which seems to be becoming more popular, suggests that the speaker enjoys a less specific sense of well-being. Any criticism of I’m good as suggesting some kind of moral superiority is misplaced. That is only one of the meanings of good and one that is clearly not intended in answer to the question How are you?
Both can be right - it isn't a specific question expecting much meaning back, but more of a general greeting.
You could also answer with "Good, thanks" or "Fine" or any number of general responses.
"I am good" has come under a lot of jovial criticism, being quasi-slang. This is more common among school-goers and pre-teens, I hear.
In proper English, it implies "I am behaving well", "I am a well-behaved person", or even "I am suitable for (whatever)".
In a while, though, the expression may gain currency and acceptance, to eventually become mainstream.
I think the question is grammatically ill-posed: "how" seems to imply that the response should be an adverb, but the full question is "how are you?", and since "you" is a noun, "well" doesn't properly modify it unless it is taken to mean "in good health."
For this reason, I generally respond with "I'm doing well," which answers the question fairly succinctly, doesn't sound terribly awkward, and is grammatically correct (since "well" modifies "doing").