Diffrerent cultures have different traditions regarding names. There is a thread here from someone in India who was trying to work out what his name would be expressed in a more European way, and because the systems are so completely different, we were not able to work out a definitive answer for him. There had to be a compromise there.
In the UK it is much the same as the US version - although of course we have lots of people from other cultures living here and their names may well follow a completely different pattern.
First/Christian name (Christian name is usually avoided now as a term, and it is not used on any official forms etc, as a lot of people are not Christians. It should only really be called a Christian name if it is your baptised name). This is everyone's everyday name, what their friends call them, etc.
Middle name(s) other names chosen by the parents. some people do not have a middle name at all, most people have one or two, some people have more.
Surname - this is your family name. In the UK, children are usually given their father's surname but sometimes they may take the mother's instead.