"Hello, this is my younger brother."
Translation:Здравствуйте, это мой младший брат.
The genitive case is for people or objects that own other things, not things that are owned.
Possessive pronouns don't change the case of the things they own, rather the possessive pronoun changes depending on the case of the noun being owned. So, if the noun was genitive case ("младшего брата"), you would use "Моего".
There are quite a few cases where you would use the genitive case,
Like the possessive: "Собака брата" - "The brother's dog" or "The dog of the brother".
Certain prepositions, "Из магазина" - "From the store", "У окна" - "By the window".
Animate(alive) masculine nouns also borrow it for the accusative. "Я вижу брата" - "I see the brother".
Negation: "Собаки нет" - "There is no dog", and probably more.
There isn't any difference between "My brother" and "My younger brother" or even "Brother" in terms of cases.