Learn Farsi: برادر من هر صبح دوش می‌گیرد. - My brother takes a shower every morning.

Because the word هر (every/each) is always followed by a singular noun in Farsi. Just like in English where we say 'every morning' instead of 'every mornings', Farsi uses هر صبح.

How does the verb دوش می‌گیرد translate to 'takes a shower'?

Farsi uses compound verbs a lot. 'To take a shower' is دوش گرفتن (dush gereftan), which literally means 'to take' or 'to catch' a shower. Here, the verb گرفتن is conjugated in the present tense for 'he' (my brother), becoming می‌گیرد (mi-girad).

I have learned that 'my brother' is برادرم. Why does this sentence use برادر من instead?

Both are correct! برادرم (barādaram) uses a possessive ending and is very common in spoken Farsi. برادر من (barādar-e man) uses the ezāfe (an invisible 'e' sound) to connect 'brother' and 'me'. It is slightly more formal or adds a bit of emphasis to the word 'my'.

Could I move هر صبح to the very end of the sentence, like we do in English?

No, in Farsi, the verb almost always goes at the very end. The compound verb دوش می‌گیرد must be the final thing in the sentence. Time phrases like هر صبح usually go right after the subject or at the very beginning of the sentence.