Qui asks about a person and can be either the subject or object of the question: qui parle? (who is speaking?), tu vois qui? (who do you see?). Que (or qu'est-ce que before a subject + verb) asks about a thing as the object: que fais-tu? or qu'est-ce que tu fais? (what are you doing?).
Quoi is used instead of que when the question word comes at the very end of a sentence or stands alone, and always after a preposition: tu fais quoi? (what are you doing? — casual), avec quoi? (with what?). Choosing between qui, que, and quoi comes down to whether the question is about a person or a thing, and where in the sentence the question word falls.