Since you specifically said that you were not asking about the use of "what" in questions I think it's safe to assume you are asking about the use of "what" and "that" as relative pronouns. I could give a very long answer about relative pronouns in general but in the interests of simplicity I will give you two basic rules about the uses of "what" and "that".
"What" refers to things, qualities, or ideas, never to people or animals. For example, "I know what happened to your bicycle" or, "The expert knew what he was talking about".
"That" refers to people, animals, things, ideas and qualities. For example, "The woman that witnessed the crime died two days later" or, "The man found the cave that contained the bear".
If you are still confused about this I'd be happy to talk about it next time you are on Skype. It's actually quite a complex subject so I can imagine you might need some extra help :)
Hasta luego :)