When you talk about your family, you constantly need to say my father, my mother, my brother, your sister. That is exactly what possessive pronouns do in German: mein, dein, sein, ihr, unser, euer. But they work in a fundamentally different way than in many other languages, so let's build the idea step by step.
In some languages, possession is shown by attaching an ending directly onto the noun itself, so the possessor and the noun fuse into one word. German never does this. The possessive is always a separate word placed in front of the noun, as in mein Vater (my father, literally "my father" as two words). Keep this first rule firmly in mind: in German it is always two separate words.
The second, trickier point is that a German possessive actually has two parts. The first part is the stem, and it depends on who the owner is: I gives mein, you gives dein, he gives sein, she gives ihr, we gives unser, you-plural gives euer. The second part is the ending, and this is the part that trips learners up, because the ending changes according to the gender and case of the thing owned — the noun that follows — not according to the owner.
For example: mein Vater has no ending at all, because Vater is masculine. But meine Mutter adds -e, because Mutter is feminine. Notice that mein itself did not change because the speaker is still "I" — what changed is the gender of the object owned. And if that masculine noun becomes the direct object (accusative case), for instance after a verb like "to love," the ending changes again to -en: Ich liebe meinen Bruder.
One detail worth flagging: the word ihr covers three different meanings depending on context — her, their, or even your (plural, informal). There is no separate form to distinguish them; only the full sentence tells you which one is meant.
A very common mistake is forgetting the ending altogether and saying mein Mutter instead of meine Mutter, because the possessive feels like it "shouldn't" change. The fix: always check the gender of the thing owned first, then pick the ending that matches it — never the gender of the owner.