Learning English grammar rules regarding capitalisation of letters is difficult for young English learners and non-native speakers alike. Many of these rules require memorisation combined with writing practice to get them right. Others are even more complex because the context of the situation written about changes whether a letter is capitalised or not; rules regarding capitalising the names of family members fall into the latter category.
According to Purdue University, proper nouns are the names of specific people, places, organisations, and sometimes things. All proper nouns in English must be capitalised, including the full names of family members. To describe one's aunt Susan, the letter "s" in Susan must always be capitalised. Likewise, to describe one's family using the dominant last name, such as Smith, the "S" in "the Smith family" must be capitalised.
When family relationships are used as proper names, they too should be capitalised. Family relationships are used as proper names when the word indicating relation (such as aunt, uncle, etc.) is used before a person's name. Therefore, if referring to "Aunt Holly" and not any other aunt, the "a" in "Aunt Holly" should be capitalised. Note that when speaking of aunts in general or when speaking of "my aunt," the letter "a" is not capitalised. Family relationships are also used as proper names when the word used indicates a specific person. If, in getting one's mother's attention, "Mother!" is called out, the word "Mother" is a proper noun that can only refer to her. "Mother," in this case is used to personally address the individual. In this case, then, the "m" in "Mother" is capitalised. The "f" in "Father," and the "g" in "Grandmother" or "Grandfather" would likewise be capitalised if used as proper nouns.
When family relationships are used with possessive pronouns, they are never capitalised. According to Purdue University, possessive pronouns are pronouns that express ownership. In English, the possessive pronouns are my, your, his, her, our, their, and whose. When a family relationship is used in conjunction with these pronouns, the first letter is never capitalised. So, when referring to "my mother" or "your aunt," the letters "m" and "a" in "mother" and "aunt" are not capitalised. Even when used with a proper name, as in Section 2 above, the family relationship is never capitalised when used with a possessive pronoun; this rule trumps all others. Therefore, if talking of "my cousin Chris" or "her sister Erin," the proper name is still capitalised (as in "Chris" and "Erin") but the word showing family relationship is not.