Learn Polish: Mój brat rozumie po polsku, ale nie mówi po polsku. - My brother understands Polish, but does not speak Polish.

When you talk about speaking, understanding, or reading a language in Polish, you use the word po followed by a special form ending in -u. So po polsku translates more closely to "in the Polish way". The word polski on its own is an adjective meaning "Polish" (like in a "Polish passport") and isn't used for the language you speak.

Why does rozumie end in -e but mówi ends in -i if they are both actions done by the brother?

Polish verbs are split into different groups (called conjugations) that follow slightly different ending rules. The verb rozumieć (to understand) belongs to a group where the "he/she/it" form ends in -e (rozumie). The verb mówić (to speak) is in a different group where the "he/she/it" form ends in -i (mówi).

Where does the word nie go when making a negative statement like "does not speak"?

In Polish, the negative word nie must be placed immediately before the verb it is negating. That is why it sits right next to the verb in nie mówi (does not speak). You don't need a helper verb like "does" in English.

Would I still use mój if the sentence was about my sister instead of my brother?