Here in New Zealand, I think that it is pretty much the same as in Britain and the US. We have a first name, which is given by our parents, and our surname, which is usually your father's surname. We also often have 'middle names'. One is common, but some people have none, and some have 2 or more. Our middle name is also given by our parents, and is often the same as the first name of a friend or older relative of one of your parents. Your middle name is only usually used for legal purposes, such as driving license, or school roll. There is no rule about your surname, so although it is usually your fathers, it can be your mothers, or even sometimes a hyphenated version of your parents, e.g. dos Santos-Pinheiro.
We are usually called by our first name, but some older people think that that is rude, and prefer to be called Mr/Mrs Surname.