My is usually classed as a possessive determiner (or possessive adjective): a word that indicates possession (telling you whom or what something or someone belongs to) by modifying the following noun (e.g., “my cat”).
It’s normally not considered a pronoun because it doesn’t stand alone in place of a noun. But it is closely related to the possessive pronoun mine, and some grammars class my and other possessive determiners as “weak possessive pronouns.”
However, it’s more useful to define my as a determiner, because this more accurately describes its role in a sentence.