Learn Hardcore Russian: Мама готовит картофель.

Мама is a noun in the feminine gender, singular, and nominative case (the subject).
Готовит is a verb in the 3rd person singular, present tense, imperfective aspect.
Картофель is a noun in the masculine gender, singular, and accusative case (the direct object).

Why is мама in the form “мама” and not “маму”?

Because мама is the subject of the sentence and thus takes the nominative case. “Маму” would be the accusative case (used for a direct object), but here мама is doing the cooking, not being cooked.

What case is картофель in, and how can you tell?

Картофель is in the accusative case, marking the direct object of the verb готовит. For inanimate masculine nouns, the accusative form is identical to the nominative form, so картофель remains unchanged.

What is the aspect of the verb готовит, and what’s its perfective counterpart?

Готовит is imperfective, indicating a process or habitual action (“is cooking” / “cooks”). The perfective counterpart is приготовит, which focuses on the completion of the action (“will cook [to completion]”).