Learn Russian: Мама лечит сына дома.

Because the son is the direct object receiving the action, the noun is in the accusative case. For animate masculine nouns like сын (words for people and animals), the accusative ending is the same as the genitive ending, which adds an -а.

The translation says "her son", but I don't see the word "her" in the Russian sentence. Where did it go?

Russian often drops possessive pronouns like её (her) or своего (one's own) when the relationship is obvious from context. Since the mother is treating a son, it is naturally understood to be her own.

Why do we use дома for "at home" instead of saying в доме?

The word дома is a special adverb that specifically means "at home" (being at your own residence). If you used в доме, it would sound like you were emphasizing being physically "inside the house" as a building, rather than just being at home.

What type of verb is лечит?

It is the "he/she/it" form of the verb лечить (to treat or cure). Because the infinitive ends in -ить, it follows the second conjugation pattern, which is why it takes the -ит ending instead of the -ет ending.