Prior to the arrival of the Pakeha, Maori did not use surnames. However, because of an oral society where whakapapa was paramount, it was the norm for people to know the relationships or everyone within the hapu and iwi.
Names were given at birth, often due to an event or circumstances. Sometimes, later on in life, a person was given another name, again, usually due to events or circumstances. For example, Tamatea Ure Haea was also known as Tamatea Pokai Whenua and Tamatea Pokai Moana.
Ngati, means descendants of, so Ngati Tuwharetoa are the descendants of Tuwharetoa. Be aware that sometimes when talking iwi and whakapapa to a whakapapa person, if you mention, in this example, Tuwharetoa, they will think you are talking about the person. If you are talking about the iwi, get into the habit of using the Ngati in front.