Each Indian tribe in the United States has the right to determine how it will define eligibility for membership. Therefore, there is no one correct answer other than to say that membership is based upon whatever a given tribe wants it to be based on.
That said, the answer that you are probably supposed to give is that membership in a tribe is based on ancestry. The majority of Indian tribes require that a person be in some degree a descendant of members of that tribe. This is usually referred to as a "blood quantum" rule. Many tribes require that the person applying for membership must be at least 1/4 Indian. That is, at least one of the person's grandparents must have been a member of that particular tribe.