Qui and que both translate as "who", "which", or "that" in English, but French chooses between them based on grammatical role, not on whether the antecedent is a person. Qui is the subject of the clause that follows it: la femme qui parle est ma mère (the woman who is speaking is my mother) — qui stands in for la femme as the subject of parle.
Que is the direct object of the clause that follows it, and it always precedes a subject + verb: le livre que je lis est passionnant (the book that I'm reading is fascinating) — que stands in for le livre as the object of lis. Que shortens to qu' before a vowel; qui never elides.