By classification, one can only assume you mean the different kind of people that might be in your family - so let's start with the basics. Firstly you have your mother and your father. Your mother is the person that gave birth to you, and whom provided the egg for you to become a living person. This is your biological mother and that can never be changed. Your biological father is the individual who provided the sperm for you to become a living person. This sperm won the race and penetrated the egg, which formed the embryo, which later became you. This is your biological father.
Some people have mothers and fathers who are not their biological mothers and fathers, however. Mothers and fathers can be the people that raised a person when their biological parents were not present in their lives. Some people love them just like they are their parents, and hence, they are considered their parents in the eyes of society.
Then you have brothers and sisters. These, biologically, are the people that were born under the same biological parents as you - though they could be people that were adopted by the same people as you, or people who only share one parent with you. So there are a few kinds of siblings.
You then have grandparents, and these are the parents of your parents, and are generally much older than you (but this of course depends on how young they were when they had your parents). You will also have uncles and aunties. These are the brothers and sisters of your parents. Though many people have other important figures in their lives that they consider aunties and uncles. You can also have cousins, who are the children of your aunties and your uncles.
Some people have mothers and fathers who are not their biological mothers and fathers, however. Mothers and fathers can be the people that raised a person when their biological parents were not present in their lives. Some people love them just like they are their parents, and hence, they are considered their parents in the eyes of society.
Then you have brothers and sisters. These, biologically, are the people that were born under the same biological parents as you - though they could be people that were adopted by the same people as you, or people who only share one parent with you. So there are a few kinds of siblings.
You then have grandparents, and these are the parents of your parents, and are generally much older than you (but this of course depends on how young they were when they had your parents). You will also have uncles and aunties. These are the brothers and sisters of your parents. Though many people have other important figures in their lives that they consider aunties and uncles. You can also have cousins, who are the children of your aunties and your uncles.
