[noun] + 或者 + [noun]
[verb phrase] + 或者 + [verb phrase]
或者 + [clause],或者 + [clause]
或者 means "or" in a statement. It joins nouns, verb phrases or whole clauses, and puts the options forward as all acceptable.
还是 asks the listener to pick one option. 或者 lays options out without asking for a choice. A 吗 question with 或者 asks whether any of them will do, and the answer is yes or no.
The mistake is 或者 in a choice question, where English uses the same "or" for both.
Repeating 或者 sets out two options more deliberately, "either … or".
或 on its own is the written form, seen in notices and forms.