Learn Hardcore Russian: Тебе нужно ответить на письмо до обеда. - You need to reply to the letter before lunch.

Тебе is the dative form of ты. In the impersonal construction with нужно, the person who needs to do something is put into the dative case. Literally, “to you it is necessary to answer the letter.”

Why is нужно used here, and how does it compare to надо, должен, or следует?

All these words express necessity or obligation, but with slight nuances:

нужно is neutral and impersonal, feels a bit softer than надо.
надо is more colloquial and also impersonal.
должен agrees with the subject (“Ты должен…”) and can sound stronger or more formal.
следует is more formal and often found in written advice.
Here, Тебе нужно… simply means “You need to…” in a neutral, everyday tone.

Why is ответить in the infinitive, and why choose the perfective aspect here?

The infinitive (ответить) follows нужно to show the action that is necessary. It is in the perfective aspect because the speaker wants to emphasize completing the action (to give a finished reply) before lunch. If you used the imperfective отвечать, it would focus on the process of replying rather than the completion.

Why is there на before , and what case does take?