Is "her" a possessive pronoun or an object pronoun or does it depend on the sentence it is used in? What are some examples?
It can be both possessive or object pronoun. See explanation for examples.
Pronoun her can be a possessive pronoun like in sentence:
Her house is very big.
But it can be also an object pronoun like in this sentence:
I cannot find Ann. Have you seen her anywhere?
In the first sentence her shows ownership. You can ask a question:
Whose house is big?
In the second sentence her is an object (a direct object to be specific). You can ask an object question:
Whom didn't you see?