Learn Russian: Мама поцеловала сына.

Поцеловала is a perfective verb, which shows that the action was a single, completed event with a clear result. If the sentence used the imperfective form, целовала, it would mean she was kissing him over a period of time or used to kiss him regularly. Just like with the time expressions in this lesson, your choice of aspect depends on whether you are focusing on a completed result or an ongoing duration!

Why does the word for son change to сына?

The son is the direct object receiving the action of the kiss, so the noun must be in the Accusative case. Because сын is a masculine animate noun (a living person), its Accusative form borrows the Genitive ending by adding -а. This is a vital rule to remember for all male people and animals!

The English translation says 'her son', but there is no word for 'her' in the Russian sentence. Why?

In Russian, possessive pronouns like 'her' (её or своего) are often dropped when the context makes the relationship obvious. Because the mother is the subject of the sentence doing the kissing, Russian naturally assumes she is kissing her own son. Adding the pronoun here wouldn't be wrong, but it is unnecessary.