Good is an adjective and well is an adverb.
Good modifies a noun; something can be or seem good.
Well modifies a verb; an action can be done well.
However, when you’re talking about health, well can be used as an adjective.
Rule: When referring to health, always use well.
Examples:
I do not feel well today.
You do not look well.
Rule: When describing someone’s emotional state, use good.
Example: He doesn’t feel good about his divorce/about losing his job/about failing the exam.
So, how should you answer the question, “How are you?”
If you think someone is asking about your physical well-being/your physical health, answer, “I feel well,” or “I don’t feel well.”
If someone is asking about your emotional state/emotional health, answer, “I feel good,” or “I don’t feel good.”
NOTE:
Nowadays, some people believe that saying "I am good" is acceptable when referring to one's health - and others think that saying "good" is incorrect and saying "well" is correct.