Luckily for me a quick Google search helped me remember what I already knew and cleared away the confusion.
The simple answer is Who is for people. That is for objects.
The more complicated answer is addressed by my favorite Grammarian, Grammar Girl (post can be found here). She found that according to the Heritage Dictionary, “It is entirely acceptable to write either the man that wanted to talk to you, or the man who wanted to talk to you.”
Grammar Girl determined it is a bit of a grey area. “My guess is that most people who use who and that interchangeably do it because they don’t know the difference,” she wrote. “To me, using that when you are talking about a person makes them seem less than human.”
Something else I learned in her post, “whose, which is the possessive form of who, to refer to both people and things because English doesn’t have a possessive form of that. it’s fine to say, ‘The desk whose top is cluttered with grammar books,’ even though it is obviously ridiculous to say, ‘The desk who is made of cherry wood.’” I didn’t know that!
Bottom line: who is for people. That is for objects.