Learn French: Il m'a expliqué son erreur. - He explained his mistake to me.

In French, object pronouns usually go right before the conjugated verb. In the passé composé, the conjugated verb is the helping verb (like a here, from avoir). So, the pronoun me is placed immediately before a, not before the past participle expliqué.

What exactly does m' stand for in this sentence?

It stands for me, which acts as an indirect object pronoun meaning "to me". Because the helping verb a starts with a vowel, the e in me is dropped (elided) to make m'a.

If erreur is a feminine noun, why does the sentence use the masculine son instead of sa?

You're right that erreur is feminine (une erreur). However, because it starts with a vowel, French uses the masculine possessive adjectives (mon, ton, son) instead of the feminine ones (ma, ta, sa). This is simply to make pronunciation smoother, avoiding the awkward sound of "sa erreur".

Do we need to make expliqué agree with the feminine noun erreur or the pronoun m'?