N2
それとも
Or, Or else
今日は文法の勉強をしますか。それとも漢字の勉強をしますか。
Will you study grammar today, or kanji?
Explanation
It starts the second question and means 'or'. After offering one choice, you give the other, and the listener picks one: 「コーヒーにしますか。それとも紅茶にしますか」 (Would you like coffee, or tea?). It is neutral and used in both casual talk and polite speech.
Mixed up with
「それとも」 is only for questions offering a choice (with か or question intonation); not in statements: not 「×明日は雨、それとも雪だ」.
「または/あるいは」 is mostly for statements and official papers, listing allowed options: 「ペンまたは万年筆で記入すること」 (Fill in with a ballpoint or a fountain pen).
When asking someone to choose, 「それとも」 is the natural word.