Learn Russian: Папа должен обещать маме купить торт.

Even though папа ends in '-а' (which is typical for feminine nouns), it refers to a male person, making it a masculine noun. The modal word for 'must' or 'have to' always matches the gender of the subject, so we use the masculine должен. If the subject were мама, it would be мама должна.

Why are there two infinitives (обещать and купить) in a row?

You can chain infinitives in Russian just like you do in English! The modal word должен requires an infinitive after it, giving us должен обещать (must promise). Then, the verb 'to promise' also takes an infinitive to express what is being promised, resulting in обещать купить (promise to buy).

Why is 'mom' written as маме here?

Маме is in the dative case. In Russian, when you promise something to someone, the person receiving the promise is an indirect object and takes the dative case. You can think of it as meaning 'must promise TO mom'.

Is the word торт in the nominative case here?

It looks exactly like the nominative case, but it is actually in the accusative case. It is the direct object of the action (the thing being bought). Because торт is a masculine inanimate noun (a non-living thing), its accusative form does not change and looks identical to its dictionary form.