Learn Polish: Ten długi tekst został napisany przez moją siostrę.

To show who performed an action in a passive sentence, Polish uses the preposition przez followed by the Accusative case. So, the Nominative moja siostra changes to the Accusative moją siostrę.

Why is the auxiliary verb został used here instead of był?

Został is used to form the passive voice for perfective (completed) verbs. Since napisać (to write) is perfective, meaning the writing of the text was finished, we use został napisany. If the action was imperfective (ongoing or repeated), we would use był pisany.

Why does the passive participle napisany end in -y?

A passive participle acts like an adjective and must agree in gender, number, and case with the subject of the sentence. Here, the subject is tekst, which is a masculine noun in the Nominative case, so napisany takes the masculine singular ending -y.

If I wanted to change this sentence back to the active voice, what happens to the cases?

In the active voice, the person doing the action becomes the Nominative subject, and the thing receiving the action becomes the Accusative object. The sentence would be: Moja siostra napisała ten długi tekst. Notice how tekst looks the same in both cases because it is an inanimate masculine noun!