いったい（一体）
…on earth, …in the world (emphasizing a question)machine translation
Formationいったい + interrogative sentence (誰が／何が／どうして…)
いったい goes before a question to EMPHASIZE the asker's bafflement, surprise, or irritation: いったい誰がこの窓を割ったんですか (who on earth broke this window?). Without いったい the sentence is still grammatical — adding it adds the feeling of "I really cannot understand this."
A mandatory condition: what follows must contain a QUESTION WORD (誰・何・どこ・どうして・いつ…). いったい latches onto that question word, so it also works in indirect questions: いったい何が起きたのか、誰も知らない (nobody knows what on earth happened).
The nuance usually leans toward annoyance or astonishment, so think carefully before using it with superiors — いったいどうしたんですか in a gentle tone is sincere concern, but in a sharp tone it becomes an interrogation. A related exclamatory word: まさか ("no way") expresses disbelief at an EVENT, while いったい attaches only to QUESTIONS.
Examples
いったい誰がこの窓を割ったんですか。
Who on earth broke this window?
こんな夜中に、いったいどこへ行くんですか。
Where on earth are you going in the middle of the night like this?
いったい何が起きたのか、誰も知りませんでした。
No one knew what on earth had happened.