Learn Russian: Повторите вопрос, пожалуйста.

This is the polite or plural command form (the imperative). Because you are making a polite request, which is also signaled by the word пожалуйста, you use the formal вы form of the verb. If you were asking a close friend or family member (ты) to repeat something, you would drop the -те and just say повтори.

Why do we use the -ите ending here instead of -йте?

The choice between -ите and -йте depends on the verb's stem. To make a command, you find the stem by dropping the ending from the "they" form of the verb (повторят becomes повтор-). Because the stem повтор- ends in a consonant, we add -ите. If the stem ended in a vowel (like чита- from читать), we would use -йте instead (читайте).

What case is вопрос in, and why didn't its ending change?

You are right to notice that вопрос is the direct object of the verb (the thing being repeated), so it must be in the Accusative case. However, вопрос is a masculine, inanimate noun (a non-living thing). A great rule to remember from earlier lessons is that masculine inanimate nouns in the Accusative case look exactly the same as their dictionary (Nominative) form.

Does пожалуйста always have to go at the end of the sentence?